Flipping in the Housing Market∗

Charles Ka Yui Leung†and Chung-Yi Tse‡ §

April 19, 2016

Abstract

We add arbitraging middlemen — investors who attempt to profit from buy-

ing low and selling high — to a canonical housing market search model. Flipping

tends to take place in sluggish and tight, but not in moderate, markets. To

follow is the possibility of multiple equilibria. In one equilibrium, most, if not

all, transactions are intermediated, resulting in rapid turnover, a high vacancy

rate, and high housing prices. In another equilibrium, few houses are bought

and sold by middlemen. Turnover is slow, few houses are vacant, and prices

are moderate. Moreover, flippers can enter and exit en masse in response to

the smallest interest rate shock. The housing market can then be intrinsically

unstable even when all flippers are akin to the arbitraging middlemen in clas-

sical finance theory. In speeding up turnover, the flipping that takes place in

a sluggish and illiquid market tends to be socially beneficial. The flipping that

takes place in a tight and liquid market can be wasteful as the efficiency gain

from any faster turnover is unlikely to be large enough to offset the loss from

more houses being left vacant in the hands of flippers.

Key words: Search and matching, housing market, liquidity, flippers

JEL classifications: D83, R30, G12

∗A previous version of the paper was circulated under the title, “Flippers in Housing Market

Search”. We thank Enrique Schroth as discussant of the paper in the 2011 NYU Economics PhD

Alumni Conference, Angela Mak for excellent research assistance and Kuang Liang Chang, Vikas

Kakkar, Fred Kwan, Edward Tang, Min Hwang, and Isabel Yan for discussions. Tse gratefully

acknowledges financial support from HK GRF grant HKU 751909H.†City University of Hong Kong‡University of Hong Kong§Corresponding author. Address: School of Economics and Finance, University of Hong Kong,

Pokfulam Road, Hong Kong. E-mail: tsechung@econ.hku.hk.

1

1 Introduction

In many housing markets, the purchases of owner-occupied houses by investors who

attempt to profit from buying low and selling high rather than for occupation are

commonplace. For a long time, anecdotal evidence abounds as to how the presence of

these investors, who are popularly known as flippers in the U.S., in the housing market

can be widespread. More recently, empirical studies have began to systematically

document the extent to which transactions in the housing market are motivated by

buying and selling for short-term gains and how these activities are correlated with

the housing price cycle. In particular, Haughwout et al. (2011) report that the share

of all new purchase mortgages in the U.S. taken out by investors — individuals who

hold two or more first-lien mortgages — was as high as 25% on average during the

early to mid 2000s. At the peak of the housing market boom in 2006, the figures

reached 35% for the whole of the U.S. and 45% for the“bubble states”. Depken et

al. (2009) report that for the same period, on average, 13.7% of housing market

transactions were for houses sold again within the first two years of purchase in the

metropolitan Las Vegas area. At the peak in 2005, it reached a high of 25%. Bayer

et al. (2011) report that for five counties in the LA metropolitan area, over 15% of

all homes purchased near the peak of the housing market boom in 2003-2005 were re-

sold within two years. Even in the cold period in the 1990s, the percentage remained

above 5%.

Arguably, the central questions on flipping in the housing market are how it may

contribute to housing price volatility and whether it serves any useful purpose. In this

paper, we study a housing market search model along the lines of Arnold (1989) and

Wheaton (1990) in which houses are demanded by flippers in addition to end-user

households to address the two questions.

In our model, the end-user households are liquidity constrained to the extent that

each cannot hold more than one house at a time. In this case, a household which

desires to move because the old house is no longer a good match must first sell it

before the household can buy a new house. The primary advantage of flippers in our

model is that, collectively, they can hold as many houses as the market needs them to

do so. A mismatched and liquidity-constrained household can then sell the old house

quickly to flippers to be able to buy a new house sooner. Thus, first of all, the flippers

in our model are akin to the arbitraging middlemen in illiquid markets in classical

finance theory, who help speed up turnover and improve liquidity in the market. Such

liquidity services, not surprisingly, are in greatest demand in an otherwise sluggish

and illiquid market in which it can take a long time for mismatched households to

sell houses themselves.

Often times, investors in the housing market are cash-rich investors as well as

experienced flippers. As cash-rich investors, they tend to have lower opportunity

costs than others in holding vacant houses.1 As experienced flippers, they should

1Helbert et al. (2013) report that many investors in the housing market are all-cash buyers.

2

be more adept at bargaining than many end-user households, who only buy and sell

houses infrequently.2 To capture such financing and bargaining advantages, in our

model, we allow for the possibility that the flippers finance real estate investment

at a lower cost and possess a greater bargaining strength than others. With these

advantages, flippers sell houses at a relatively high price, in which case mismatched

households can be better off letting flippers sell on their behalf, irrespective of how

quickly they can sell houses themselves, if the flippers are not buying houses from

them at too big a discount. In a tight and liquid market where houses are sold

quickly, there can only be a small “bid-ask” spread in house flipping. Then, flippers

must be buying houses from mismatched households at a similarly high price if they

are selling those houses later on at a high price. The main novelty in our analysis

is that we find that mismatched households may find it attractive to sell to flippers

not just in sluggish and illiquid markets, but also in tight and liquid markets for

the especially high price they receive from selling to flippers in such markets where

the flipper-sellers’ advantages are passed onto household-sellers to the fullest extent

possible.

Because flippers can also thrive in a tight and liquid market while the market tends

to be tight and liquid when flipping is prevalent, there can be multiple equilibria in our

model. In one equilibrium, most, if not all, transactions are intermediated, resulting

in rapid turnover and high housing prices. In another equilibrium, few houses are

bought and sold by flippers. Turnover is slow and prices are moderate.

With the multiplicity of equilibrium, wide swings in price and transaction can

happen without any underlying changes in housing supply, preference, and interest

rate. Moreover, in our model, flippers can enter and exit en masse in response to

the smallest interest rate shock. Then, on top of the usual effect of interest rates

on asset prices, any such shocks can have a significant indirect impact on housing

prices through their influences on the activities of flippers. In all, we find that even

in the entire absence of any kind of extrinsic uncertainty and less-than-fully rational

agents, flipping can still contribute to housing price volatility. A natural question

to follow up is how important such a channel of volatility can be. Our quantitative

analysis indicates that housing prices can differ by up to 23 percent across steady-state

equilibria and vary by 26 percent in response to a seemingly unimportant interest rate

shock when the model is calibrated to several observable characteristics of the U.S.

housing market.

While more houses are being flipped and remain vacant in the hands of flippers,

more households have to seek shelter in rental housing. In the model housing market,

flipping can be excessive if the efficiency gains from the faster turnover fall short of

the rental expenditures households incur during which houses are being flipped and

left vacant. In an equilibrium where mismatched households are selling to flippers

primarily for the high price flippers offer, they do not tend to be better off than if

2Bayer et al. (2011) document the prevalence of experienced flippers in the housing market and

show that they often buy houses at lower prices and sell them at higher prices than others.

3

there were no flipping, for later on, the households will be buying new houses at a

similarly high price. In this case, there is little to gain from flipping to offset its cost

and the strategic complementary that gives rise to flipping in tight markets can be a

form of market failure.

Our model has a number of readily testable implications. First, it trivially predicts

a positive cross-section relation between housing prices and Time-On-The-Market

(TOM) — mismatched homeowners can either sell quickly to flippers at a discount

or to wait for a better offer from an end-user buyer to arrive — which agrees with

the evidence reported in Merlo and Ortalo-Magne (2004), Leung et al. (2002) and

Genesove and Mayer (1997), among others.3

An important goal of the recent housing market search and matching literature

is to understand the positive time-series correlation between housing prices and sales

and the negative correlation between the two and the average TOM.4 In our model,

across steady-state equilibria, a positive relation between prices and sales and a neg-

ative relation between the two and the average TOM also hold — in the equilibrium

in which more houses are sold to flippers, prices and sales are both higher, whereas

houses on average stay on the market for a shorter period of time. More importantly,

our analysis adds a new twist to the time-series empirics of the housing market, which

is that vacancies should increase together with prices and sales if the increase in sales

is due to more houses sold to flippers, who will just leave them vacant until they are

sold to the eventual end users.

Insofar as the flippers in our model act as middlemen between the original home-

owners and the eventual end-user buyers, this paper contributes to the literature on

middlemen in search and matching pioneered by Rubinstein and Wolinsky (1987).

Previously, it was argued that middlemen could survive by developing reputations

as sellers of high quality goods (Li, 1998), by holding a large inventory of differ-

entiated products to make shopping less costly for others (Johri and Leach, 2002;

Shevchenko, 2004; Smith, 2004), by raising the matching rate in case matching is

subject to increasing returns (Masters, 2007), and by lowering distance-related trade

costs for others (Tse, 2011). This paper studies the role of middlemen in the provision

of market liquidity and the effects of any financing and bargaining advantages that

middlemen may possess on the nature of equilibrium.

Among papers in the housing market search and matching literature, perhaps

Moen and Nenov (2014) is closest to ours in that they study how decisions made by

mismatched households, like those in our model, can lead to multiplicity. In their

model, households are permitted to hold up to two houses, but holding two houses is

3Albrecht et al. (2007) emphasize another aspect of the results reported in Merlo and Ortalo-

Magne (2004), which is that downward price revisions are increasingly likely when a house spends

more and more time on the market.4Stein (1995), who explains how the down-payment requirement plays a crucial role in amplifying

shocks, is an early non-search-theoretic explanation for the positive relation between prices and sales.

Hort (2000) and Leung et al. (2003), among others, provide recent evidence. Kwok and Tse (2006)

show that the same relation holds in the cross section.

4

a particularly costly state, and so is holding no house. The strategic complementary

in their model is that when other mismatched households are choosing to buy a

new home first before selling, there will be a tight market, in which an individual

mismatched household is also better off to buy first before selling. To do otherwise

can leave the household in the costly state of holding no house for an extended period

of time. Conversely, when other households choose selling first, there will be a loose

market, in which it is optimal for an individual household to also choose selling first to

avoid being stuck in the costly state of holding two houses for a long time. Like their

model, decisions made by mismatched households in ours impart on market tightness,

through which the strategic complementary works. Unlike their model in which prices

do not appear to play any role in the strategic complementary, how prices vary with

market tightness is crucial to the strategic complementary in ours. It is this last

mechanism through which our model gives rise to the implication that housing prices

are positively correlated with sales and vacancies but negatively correlated with TOM

in the time series.

A simple model of housing market flippers as middlemen is also in Bayer et al.

(2011). The model though is partial equilibrium in nature and cannot be used to

answer many of the questions we ask in this paper. Intermediaries who buy up mis-

matched houses from households and then sell them on their behalf are also present

in the model of the interaction of the frictional housing and labor markets of Head

and Lloyd-Ellis (2012). But there the assumption is merely a simplifying assump-

tion and the presence of these agents in the given setting appears inconsequential.

Analyzes of how middlemen may serve to improve liquidity in a search market also

include Gavazza (2012) and Lagos et al. (2011). These studies do not allow end-user

households a choice of whether to deal with the middlemen and for the multiplicity

of equilibrium like we do though. Multiple equilibria in a search and matching model

with middlemen can also exist in Watanabe (2010). The multiplicity in that model

is due to the assumption that the intermediation technology is subject to increasing

returns to scale, whereas the multiplicity in ours arises from a particular strategic

complementary. Moreover, only one of the two steady-state equilibria in that model

is stable, whereas there can be more than one stable steady-state equilibria in ours.

The next section presents the model, the detailed analysis of which follows in Sec-

tion 3. In section 4, we study the planner’s problem of optimal flipping in the model

housing market. Section 5 explores various empirical implications of the model. In

section 6, we test the time-series implications of our model as pertain to especially the

behavior of the vacancy rate. In Section 7, we calibrate the model to several observ-

able characteristics of the U.S. housing market to assess the amount of volatility that

the model can generate. Section 8, which draws on results we present in a technical

note (Leung and Tse, 2016a) to accompany the paper, discusses several extensions

of the model. Section 9 concludes. All proofs are relegated to the Appendix. For

brevity, we restrict attention to analyzing steady-state equilibria in this paper. A

second companion technical note (Leung and Tse, 2016b) covers the analysis of the

5

dynamics for the special case in which all agents possess the same bargaining power.5

2 Model

2.1 Basics

There is a continuum of measure one risk-neutral households, each of whom discounts

the future at the rate r. There are two types of housing: owner-occupied, the supply

of which is perfectly inelastic at H < 1 and rental, which is supplied perfectly elas-

tically for a rental payment of q per time unit. A household staying in a matched

owner-occupied house enjoys a flow utility of υ > 0, whereas a household either in

a mismatched house or in rental housing none. A household-house match breaks up

exogenously at a Poisson arrival rate δ, after which the household may continue to

stay in the house but it no longer enjoys the flow utility υ. In the meantime, the

household may choose to sell the old house and search out a new match. An impor-

tant assumption is that households are liquidity constrained to the extent that each

can hold at most one house at a time. Then, a mismatched homeowner must first

sell the old house and move to rental housing before she can buy a new house. Our

qualitative results should hold as long as there is a limit, not necessarily one, on the

number of houses a household can own at a time.6 The one-house-limit assumption

simplifies the analysis considerably.

The search market Households buy and sell houses in a search market in which

the flow of matches is governed by a concave and CRS matching function M (B,S),

with B and S denoting, respectively, the measures of buyers and sellers in the market.

Let θ = B/S denote market tightness. Then, the rate at which a seller finds a buyer

is

η (θ) ≡ M (B,S)

S=M (θ, 1) ,

whereas the buyer’s matching rate is μ (θ) = η (θ) /θ. Given thatM is increasing and

concave in B and S,∂η

∂θ> 0,

∂μ

∂θ< 0.

We impose the usual regularity conditions on M to ensure that

limθ→0

η (θ) = limθ→∞

μ (θ) = 0, limθ→∞

η (θ) = limθ→0

μ (θ) =∞.5Not for publication, available for download in http://www.sef.hku.hk/~tsechung/index.htm6In Section 8 and in Leung and Tse (2016a), we explain how this is the case when households

can hold up to two houses at a time.

6

The Walrasian investment market Instead of waiting out a buyer to arrive in

the search market, a recently mismatched household may sell the old house right

away in a Walrasian market to specialist investors — agents who do not live in the

houses they have bought but rather attempt to profit from buying low and selling

high. Because homogeneous flippers do not gain by buying and selling houses among

themselves, the risk-neutral flippers may only sell in the end-user search market and

will succeed in doing so at the same rate η that any household-seller does in the

market. We assume that flippers discount the future at the same rate r that end-user

households do but allow for the possibility that they finance real estate investment at

a different rate rF . In the competitive investment market, prices adjust to eliminate

any excess returns on real estate investment.

The assumption of a Walrasian investment market is, by all means, a simplifying

assumption. A more general assumption is to model the market as a search market

too, but possibly one in which the frictions are less severe than in the end-user market.

If flippers are entirely motivated by arbitrage considerations and do not care if the

houses to be purchased are good matches for their own occupation, search should be

a much less serious problem. Moreover, if the supply of potential flippers is perfectly

elastic at a zero entry cost, in equilibrium, there will be an infinite mass of flipper-

buyers in the market. In this case, any households who wish to sell in the investment

market can do so instantaneously. That is, a Walrasian investment market can be

thought of as the limit of a search market in which the cost of entry for flippers tends

to zero.

2.2 Stocks and flows

Accounting identities At any one time, a household can either be staying in a

matched house, in a mismatched house, or in rental housing. Let nM , nU , and nRdenote the measures of households in the respective states, which must sum to the

unit measure of households in the market; i.e.,

nM + nU + nR = 1. (1)

Each owner-occupied house must be held either by an end-user household or by a

flipper. Hence,

nM + nU + nF = H, (2)

where nF denotes both the measures of active flippers and houses held by these agents.

If each household can hold at most one house at any moment, the only buyers in

the search market are households in rental housing; i.e.,

B = nR. (3)

On the other hand, sellers in the search market include both mismatched homeowners

and flippers, so that

S = nU + nF . (4)

7

Figure 1: The flows of households into and out of the three states

Housing market flows In each unit of time, the inflows into matched owner-

occupied housing are comprised of the successful buyers among all households in

rental housing (μ (θ)nR), whereas the outflows are comprised of those who become

mismatched in the same time period (δnM). In the steady state,

·nM = 0⇒ μ (θ)nR = δnM . (5)

Households’ whose matches just break up may choose to sell their old houses right

away to flippers in the investment market or to wait out a buyer to arrive in the

search market. Let α denote the (endogenously determined) fraction of mismatched

households who sell in the investment market and 1 − α the fraction who sell in

the search market. In each time unit then, the measure of mismatched homeowners

selling in the search market increases by (1− α) δnM , whereas the exits are comprised

of the successful sellers (η (θ)nU) in the meantime. In the steady state,

·nU = 0⇒ (1− α) δnM = η (θ)nU . (6)

It can be shown that (5) and (6), together with θ = nRnU+nF

and μ = η

θ, imply that

αδnM + η (θ)nU = μ (θ)nR,

αδnM = η (θ)nF ,

8

which say that, respectively, the flows into and out of rental housing are equal³ ·nR = 0

´and the measure of houses bought by flippers are matched by the mea-

sure sold³ ·nF = 0

´. Figure 1 depicts the flows of households into and out of the

three states.

2.3 Market tightness

Solving (1)-(6), we can show the following.

Lemma 1

a. At α = 0, nF = 0.

b. As α increases from 0 toward 1,

∂nF

∂α> 0,

∂nR

∂α> 0,

∂nM

∂α> 0, whereas

∂nU

∂α< 0.

c. At α = 1, nU = 0.

Parts (a) and (c) of the Lemma say that, respectively, at one extreme, when no

mismatched households are selling to flippers, there cannot be any vacant houses

in the hands of these agents, and at the other extreme, when all households sell

to flippers immediately after becoming mismatched, there cannot be any household

remaining in a mismatched house in the steady state. In general, by part (b), as α

increases from 0 toward 1, there are more houses in the hands of flippers and fewer

in the hands of mismatched homeowners.

What is less obvious in the Lemma is that as α increases, there would also be

more households staying in rental housing, as well as being matched in the steady

state. Intuitively, when flippers hold a greater fraction of the owner-occupied housing

stock, fewer households can stay in owner-occupied houses and therefore more must

be accommodated in rental housing. In the meantime, there are fewer mismatched

households spending any time at all selling their old houses in the search market

before initiating search for a new match. Then, there should only be more matched

households in the steady state.

When more mismatched households sell to flippers right away, there are not only

more buyers in the search market but also fewer sellers as well.7 This is because as

houses are sold faster where there are more buyers, there can only be a smaller stock

of houses left for sale in the steady state. The two tendencies should then conspire

to give rise to a tighter market with a larger

θ =B

S=

nR

nU + nF.

7By (2), S = nU + nF = H − nM , which is decreasing in α given that ∂nM/∂α > 0.

9

Lemma 2 The stock-flow equations (1)-(6) can be combined to yield a single equation,

δ + η (θ) (1−H)− (1− α+ θ)Hδ = 0, (7)

in α and θ, from which an implicit function θ = θS (α) for α ∈ [0, 1], can be defined,and that ∂θS/∂α > 0. Both the lower and upper bounds, given by, respectively, θS (0)

and θS (1), are strictly positive and finite.

2.4 Asset values and housing prices

Lemmas 1 and 2 above describe how the measures of households in the three states,

the stock of houses in the hands of flippers, and the market tightness depends on

α. The value of α — for the fraction of mismatched households selling to flippers —

depends on the households’ comparison of the payoffs of selling in the investment

versus the search markets.

Asset values for households A household staying in a mismatched house and

trying to sell it in the search market to an end user has asset value VU satisfying,

rVU = η (θ) (VR + pH − VU) , (8)

where pH denotes the price the household expects to receive for selling the house in

the search market to another household and VR the value of being in rental housing.

Under (8), the mismatched homeowner is entirely preoccupied with disposing the old

house while she makes no attempt to search for a new match. This is due to the

assumption that a household cannot hold more than one house at a time. Once the

household manages to sell the old house and only then, it moves to rental housing and

enters the search market as a would-be buyer, who may eventually be buying either

from a mismatched homeowner at price pH or from a flipper at a price we denote as

pFS.

Lemma 3 In the steady state, the fraction of flipper-sellers among all sellers in the

search market is equal to the fraction of mismatched households selling to flippers in

the first place; i.e.,nF

nF + nU= α.

In this case,

rVR = −q + μ (θ) (VM − (αpFS + (1− α) pH)− VR) , (9)

where q is the exogenously given flow rental payment and VM the value of staying in

a matched house, given by

rVM = υ + δ (max {VR + pFB, VU}− VM) . (10)

10

The owner-occupier enjoys the flow utility υ while being matched. The match will

be broken, however, with probability δ, after which the household may sell the house

right away in the investment market at a price we denote as pFB and switch to

rental housing immediately thereafter. Alternatively, the household can continue to

stay in the house while trying to sell it in the search market. Note that under (10)

and (8), the mismatched household has one chance only to sell the old house in

the investment market, at the moment the match is broken. Those who forfeit this

one-time opportunity must wait out a buyer in the search market to arrive. The

restriction is without loss of generality though in a steady-state equilibrium, in which

the asset values and housing prices stay unchanging over time.8

Asset values for flippers If a flipper expects to receive pFS for selling a house in

the search market and has previously paid pFB for it in the investment market, the

flipper has asset value VF satisfying,

rVF = −rFpFB + η (θ) (pFS − pFB − VF ) ,

where rF is the flipper’s cost of funds. With free entry in house flipping, VF = 0 and

therefore,

pFB =η (θ)

η (θ) + rFpFS. (11)

Bargaining When a household-buyer in rental housing is matched with a flipper-

seller, the division of surplus in the bargaining satisfies

βF (VM − pFS − VR) = (1− βF ) (pFS − pFB) , (12)

where βF denotes the flipper-seller’s share of the match surplus. When the same buyer

is matched with a household-seller in a mismatched house, the division of surplus in

the bargaining satisfies

βH (VM − pH − VR) = (1− βH) (VR + pH − VU) , (13)

where βH denotes the household-seller’s share of the match surplus. If flippers are

agents specializing in buying and selling, it is most reasonable to assume that βF ≥βH .

8Equation (9) assumes that the rental household is better off buying a house either from a

mismatched homeower or a flipper rather than continuing to stay in rental housing. Equations

(10) and (8) assume that the mismatched household is better off selling the old house either in the

investment or search market rather than just staying in the mismatched house. Lemma 13 in the

Appendix verifies that all this holds in equilibrium.

11

2.5 Households’ optimization

Write

D (θ,α) ≡ VR + pFB − VU , (14)

as the difference in payoff for a mismatched household between selling in the invest-

ment market (VR + pFB) and in the search market (VU). If D (θ,α) > 0 (< 0), for

all α ∈ [0, 1], the household strictly prefers to sell in the investment (search) marketat the given θ no matter what others choose to do. For certain θ, there may exist

some αD (θ) ∈ [0, 1] such that D (θ,αD (θ)) = 0, in which case equilibrium requires

a certain fraction αD (θ) of mismatched households selling in the investment market

and the rest selling in the search market. In sum, we can define a relation,

αO (θ) =

⎧⎨⎩ 1

αD (θ)

0

D (θ, 1) ≥ 0D (θ,αD (θ)) = 0

D (θ, 0) ≤ 0,

between market tightness θ and the fraction α of mismatched households selling in

the investment market from the households’ optimization.

2.6 Equilibrium

We now have two steady-state relations between α and θ: the θS (α) function in

Lemma 2 from the stock-flow equations and the αO (θ) relation from mismatched

households’ optimization. A steady-state equilibrium is any {α, θ} pair that simulta-neously satisfies the two relations.9

3 Analysis

3.1 Flippers’ advantages and the nature of equilibrium

To proceed with the analysis of equilibrium, we begin with the characterization of

the D (θ,α) function that underlies the αO (θ) relation. By (11)-(13),

D (θ,α) = (1− βH)−1µβHrF + βFη (θ)

βFη (θ)pFB − pH

¶. (15)

For βH 6= 1, (15) has the same sign as

pFB − pH η (θ)βFη (θ)βF + rFβH

. (16)

Recall that a mismatched household receives pFB right away if it sells in the invest-

ment market, whereas it will receive pH at some uncertain future date if it offers the

9Proposition 6 in the Appendix establishes the existence of equilibrium.

12

house for sale in the search market. By (16), the comparison in (14) is likened to a

comparison between the instantaneous reward pFB and an appropriately discounted

future reward pH of selling the mismatched house in the two markets.

In a tighter search market with a larger θ, houses, on average, are sold faster in

the market. Then, pH in (16) should be discounted less heavily in the comparison of

the payoffs between selling in the two markets. Indeed, for given pFB and pH , the

expression in (16) is decreasing in θ, whereby households should find selling in the

investment market less attractive when they can sell their mismatched houses faster

in the tighter search market.

Both pFB and pH , however, can also depend on θ. How D (θ,α) behaves as a

function of θ can only be resolved by also checking how the two prices vary with θ.

In Lemma 12 in the Appendix, we present the solutions of pFB and pH , together with

those of the various asset values, from (8)-(13). Substituting in the solutions to (15),

D (θ,α) is seen to have the same sign as

bD (θ,α) ≡ µβF rrF − βH − zβH¶η (θ) + (1− βH − α (βF − βH))μ (θ)− (δ + r) z,

(17)

where z = q/υ.10

3.1.1 Inventory advantage

It is straightforward to verify that bD (θ,α) is indeed everywhere decreasing in θ for

rF ≥ βFβH

r

1 + z≡ r. (18)

Lemma 4 For rF ≥ r, αO (θ) is everywhere non-increasing, given by,

αO (θ) =

⎧⎨⎩1 θ ≤ θd1αD (θ) θ ∈ ¡θd1, θd0¢0 θ ≥ θd0

,

where θd1 < θd0 are defined by, respectively,bD ¡θd1, 1¢ = 0 and bD ¡θd0, 0¢ = 0, and that

∂αD (θ) /∂θ < 0.11

Panel A of Figure 2 illustrates an example of the αO (θ) function in the Lemma.

Here, as expected, mismatched households only prefer selling in the investment market

10Lemma 12 in the Appendix presents two sets of prices and asset values, one derived under the

assumption that D ≤ 0 and the other D ≥ 0. In either case, D is seen to have the same sign as bDin (17).11The superscript “d” denotes the θ is at where bD is decreasing in θ. Likewise, the superscript

“u” is used later on to denote the θ is at where bD is increasing in θ.

13

Figure 2: The unique equilibrium for rF ≥ r

for small θ to avoid the possibly lengthy wait in selling in a sluggish search market.

The condition of the Lemma, by (18), is met for rF = r and βF = βH . This means

that in the absence of any financing or bargaining advantage over end-user households,

flippers may survive only when the search market is relatively illiquid due to a small

θ and on the basis of helping mismatched households overcome the liquidity problems

that they face in the market.

To pin down the equilibrium α and θ, we invert θS (α) in Lemma 2 to define

αS ≡ θ−1S , whereby αS : [θS (0) , θS (1)] → [0, 1]. In the three panels of Figure 2,

we superimpose a different αS (θ) onto the same αO (θ). Given a strictly increasing

αS (θ) and a non-increasing αO (θ), equilibrium is guaranteed unique. In Panel A, at

θ = θS (0), αO (θ) = 0; then {α, θ} = {0, θS (0)} is the unique steady-state equilib-rium. In this equilibrium, which we refer to as the no-intermediation equilibrium, all

sales and purchases are between two end users. In Panel C, at θ = θS (1), αO (θ) = 1;

then {α, θ} = {1, θS (1)} is the unique steady-state equilibrium. In this equilibrium,which we refer to as the fully-intermediated equilibrium, all transactions are inter-

mediated by flippers. In between, there can be equilibria at where bD (θS (α) ,α) = 0,as illustrated in Panel B. In such partially-intermediated equilibria, with mismatched

households indifferent between selling in the investment and search markets, a frac-

tion, but only a fraction, of all transactions are intermediated.

What distinguishes between the situations depicted in Panel A, on the one hand,

with α = 0 in equilibrium, and Panels B and C, on the other hand, with α > 0 in

equilibrium is whether or not

θS (0) > θd0, (19)

by which the smallest feasible market tightness exceeds the largest θ for which house-

holds still find it optimal to sell to flippers. In this case, there can be no flipping in

equilibrium.

The Appendix shows that limH→0 θS (α) =∞ — in case of a zero housing supply,

14

there can be no houses for sale and market tightness reaches infinity. Conversely,

limH→1 θS (0) = 0 — if there is one house for every household and if no one is sellingto flippers where α = 0, no mismatched households can rid themselves of their old

houses to start searching for a new match and market tightness falls to zero. In

general, ∂θS/∂H < 0 at any α — with a larger housing stock, more houses are for sale

and then as houses are sold more slowly as a result, there can only be fewer successful

household-sellers entering the search market as buyers, which together conspire to

loosen the market. θd0, on the RHS of (19) for the largest θ for which households still

find selling in the investment market attractive, is decreasing in rF , with a strictly

positive lower bound←−θ d0¡≡ limrF→∞ θd0

¢. All this means that, as in Figure 3, for

rF ≥ r, we can draw an upward-sloping border in the H-rF space, along which thetwo sides of (19) are just equal and to the right (left) are combinations of {H, rF} forwhich α > 0 (α = 0) in the unique equilibrium.12 Intuitively, with a larger H, there

will be a more sluggish market with a smaller θ and it becomes more attractive for

mismatched households to sell to flippers. The indifference condition bD = 0 can thenbe only restored at a lower flippers’ buy-price pFB resulting from a larger rF . Indeed,

(19) can hold in reverse, for which α > 0 in equilibrium, even for an arbitrarily large

rF if there is also a large enough H, given that limH→1 θS (0) = 0 and a strictly

positive←−θ d0. That is, there can be a liquidity provision role for flippers even when

they are handicapped by a very large financing cost if, in the meantime, there is a

large enough housing stock to make selling in the search market very difficult for

mismatched households.

3.1.2 Financing/Bargaining advantage

For all rF > 0, bD (θ,α) in (17) is at first decreasing in θ — there is less of a liquidity-

provision role for flippers as the search market gets tighter and more liquid. For

rF < r, however, which holds for rF sufficiently below r and/or for βF sufficiently

above βH ,13 bD (θ,α) is eventually increasing in θ. Altogether, bD (θ,α) is U-shaped

if moreover ∂ bD (θ,α) /∂θ changes sign just once, which is guaranteed to be the caseunder a fairly weak condition on η (θ) — a condition we assume holds in the follow-

ing.14 That is, if the flipper-seller possesses some sufficiently large financing and/or

bargaining advantages over the household-seller, there is a force, other than for liq-

uidity reason, that makes selling in the investment market attractive for mismatched

12The construction of the diagram is explained in more details in the Appendix.13Given that the condition rF < r, by (18), is a condition on by how much rF /βF is below r/βH ,

it appears that it may be more illuminating to state results in terms of the comparison of the two

ratios instead of stating results in terms of the level of rF . The conditions to follow in Lemma 5,

however, cannot be stated just in terms of one and/or both ratios. For coherence, we think it is

least confusing to state conditions throughout in terms of the level of rF .14The condition is 2∂η

∂θ

³η (θ)− θ ∂η

∂θ

´+ θ ∂

2η

∂θ2η (θ) ≤ 0, which is guaranteed to hold if η (θ) is

isoelastic.

15

Figure 3: The nature of equilibrium

households, and such a force strengthens as the market gets tighter.

Where rF < r, the flipper-seller has a lower opportunity cost of holding the

vacant house and therefore a more favorable outside option in bargaining than the

household-seller. Where βF > βH , the flipper-seller can extract a greater share of

the match surplus than the household-seller can. With one or both advantages, the

flipper-seller would be able to bargain for a price pFS above the price pH that the

mismatched homeowner can bargain for herself in the sale to the same end-user buyer,

other things being equal. In a tighter and more liquid market, by (11), the “bid-ask”

spread in house flipping narrows as houses are sold more quickly. Then, as the market

tightens, if the flipper is able to sell a house at a relatively high pFS in the search

market, the flipper must be paying a more than proportionately higher pFB for the

house earlier in the investment market. In all, for rF < r, mismatched households

may also find selling to flippers attractive for the especially high price flippers offer

when the market is tight and liquid as the flipper-seller’s advantages are passed onto

the household-seller to the fullest extent possible in such a market.

Lemma 5 There exist some r0 and r00, with 0 < r0 < r00 ≤ r, such that

a. For rF ∈ (0, r0],αO (θ) = 1 for θ ≥ 0.

16

b. For rF ∈ (r0, r00),

αO (θ) =

⎧⎨⎩ 1 θ ≤ θd1αD (θ) θ ∈ ¡θd1, θu1¢1 θ ≥ θu1

,

where θd1 < θu1 are defined bybD ¡θd1, 1¢ = 0 and bD (θu1 , 1) = 0 over wherebD (θ,α) is decreasing and increasing in θ, respectively. Here, αD (θ) is first

decreasing, reaching a minimum above zero, and then increasing toward 1.

c. For rF ∈ [r00, r) ,

(c) αO (θ) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩1 θ ≤ θd1αD (θ) θ ∈ ¡θd1,θd0¢0 θ ∈ £θd0, θu0¤αD (θ) θ ∈ (θu0 ,θu1)1 θ ≥ θu1

,

where θd1 < θd0 < θu0 < θu1 are defined by, respectively,bD ¡θd1, 1¢ = 0 andbD ¡θd0, 0¢ = 0, over where bD (θ,α) is decreasing in θ, and bD (θu0 , 0) = 0 andbD (θu1 , 1) = 0 over where bD (θ,α) is increasing in θ. For θ ∈ ¡θd1,θd0¢, ∂αD (θ) /∂θ <

0, whereas for θ ∈ (θu0 ,θu1), ∂αD (θ) /∂θ > 0.

Part (a) of the Lemma covers the situation in which the U-shaped bD (θ,α) functionstays above zero for all α and θ, meaning that the mismatched household’s payoff of

selling in the investment market exceeds that of in the search market at any market

tightness. Then, with αO (θ) = 1 for all θ, as illustrated in Panel A of Figure 4, the

unique equilibrium must be at α = 1 for any values of H, as indicated in Figure

3. In this case, flippers’ financing/bargaining advantage is so overwhelming that the

unique equilibrium must be a fully-intermediated equilibrium.

Lemma 5(b) covers the situation for rF above a first threshold r0 but below a sec-

ond threshold r00, whereby αO (θ), though can fall below one, never falls all the way

down to zero for any θ, as illustrated in Panel B of Figure 4, so that any equilibrium

must be at where α > 0, as indicated in Figure 3. In this case, flippers’ financ-

ing/bargaining advantage, though remains operative, only suffices to allow them to

offer a price attractive enough to lure all mismatched households to sell in the invest-

ment market when the search market is sufficiently tight.

As rF reaches and rises above the r00 threshold, Lemma 5(c) says that αO (θ) now

does fall to zero over a given interval£θd0, θ

u0

¤of market tightness, as illustrated in

Panel C of Figure 4. In this case, if

θd0 ≤ θS (0) ≤ θu0 , (20)

17

Figure 4: The αO (θ) function : rF < r

there indeed exists a no-intermediation equilibrium with α = 0. Otherwise, any

equilibrium must still be at where α > 0.

As rF rises toward r, flippers’ financing/bargaining advantage weakens. For-

mally, the interval£θd0, θ

u0

¤widens with ∂θd0/∂rF < 0 and ∂θu0/∂rF > 0.15 Given

that ∂θS (0) /∂H < 0, we can divide the H-rF space for rF ∈ [r00, r) in Figure 3 intothree subsets with the middle, but only the middle, subset made up of all {H, rF}for which (20) holds. That the border dividing the second and the third subsets is

upward-sloping has the same interpretation as to why the border in the Figure for

rF ≥ r is upward-sloping: an increase in rF hinders flippers’ ability to offer a priceattractive enough to allow them to carry out their liquidity-provision role so that

flipping can continue to take place only when the search market is more illiquid due

to a larger housing stock. To understand why the border dividing the first and second

subsets slopes down, note that the first subset is made up of all {H, rF} for which

θS (0) > θu0 , (21)

whereby the second inequality in (20) holds in reverse. The condition says that

the smallest feasible market tightness θS (0) is above the lower bound θu0 of market

tightness above which the financing/bargaining advantage of flippers would allow

them to lure at least a fraction of mismatched households to sell in the investment

market. As rF increases, the advantage weakens and the lower bound θu0 increases in

value. For (21) to continue to hold, there has to be a smaller H to cause θS (0) to go

up in tandem.

15Indeed, limrF→r θu0 =∞, whereby the αO (θ) function in Panel C of Figure 4 turns into the one

in Figure 2.

18

3.1.3 Summing up

In the model housing market, flippers possess two advantages over end-user house-

holds: (1) inventory advantage and (2) financing/bargaining advantage. For the

smallest rF and largest βF , the financing/bargaining advantage suffices to enable

flippers to offer a good enough price to lure all mismatched households to sell in the

investment market at any level of market tightness in the search market resulting

from any level of housing supply. As the advantage weakens, flippers may only sur-

vive in sluggish markets on the basis of helping mismatched households overcome the

liquidity problems and in tight markets on the basis of offering a very attractive price.

Finally, when the financing/bargaining advantage weakens further and disappears al-

together, flippers may only survive on the basis of providing liquidity services. In this

case, equilibrium is guaranteed unique and may involve flipping only for a relatively

sluggish market resulting from a large housing supply.

3.2 Multiplicity

In Panels B and C of Figure 4, where αO (θ) becomes an increasing function over a

range of θ, there can well be multiple equilibria given that αS (θ) is upward sloping

throughout. Figures 5 and 6 illustrate two examples of multiplicity. In both examples,

there are as many as three equilibria. Full intermediation, with α = 1, is equilibrium

in the examples since at θ = θS (1), flippers will be paying a price attractive enough

to lure all mismatched households to sell in the investment market, while if all mis-

matched houses are sold in the investment market right away, the rapid turnover will

indeed give rise to a tight market with θ = θS (1). In a slower market, flippers will

no longer be paying a pFB attractive enough to lure all mismatched households to

sell in the investment market. But precisely because fewer or none at all mismatched

houses are sold in the investment market, a relatively sluggish search market will

emerge from the slower turnover. As a result, a smaller α and a smaller θ is also

equilibrium in Figures 5 and 6.

Consider a small perturbation from the middle equilibrium in Figures 5 and 6 that

knocks the {α, θ} pair off to the right of the αO (θ) function. Then, bD (θ,α) > 0 since∂ bD/∂θ > 0 for θ ≥ θu0 , after which all mismatched households will find it better to sell

in the investment market. The increase in turnover will raise θ further. Eventually the

market should settle at the α = 1 equilibrium. Conversely, a perturbation that knocks

the {α, θ} pair off to the left of αO (θ) function from the middle equilibrium in Figures5 and 6 should send the market to a smaller α equilibrium. In general, an equilibrium

at where αO (θ) is increasing should be unstable. By analogous arguments, the other

equilibria in the two examples should be locally stable.16 Hence, there are not just

multiple steady-state equilibria but also multiple locally stable steady-state equilibria.

A necessary condition for multiplicity, by Lemmas 4 and 5, is that rF ∈ (r0, r)16A more rigorous local stability analysis is in Leung and Tse (2016b).

19

Figure 5: Multiple Equilibria: θd0 > θS (0)

Figure 6: Mutliple Equilibria: θS (0) > θd0

20

over which αO (θ) becomes increasing over a range of θ. The two examples above

suggest that for rF ∈ [r00, r), suffice for the existence of multiple equilibria is thatθS (0) < θu0 < θu1 ≤ θS (1) , (22)

where the first inequality ensures the existence of at least one α < 1 equilibrium and

the last inequality the existence of the α = 1 equilibrium.

Lemma 6 Assume that η (θ) = aθb for some a > 0 and b ∈ (0, 1). The firstinequality of (23) is the condition for rF > r

00, whereas the second inequality and (24)together guarantee that θu1 < 1 and rF < r.

rβFµ³b

1−βH

´b(δ + r) z

a

¶ 11−b(1− b) + (1 + z)βH

< rF (23)

<rβF

(δ + r) za− (1− βF ) + (1 + z)βH

.

(δ + r)z

a≥ (1− βF ) , (24)

With θS (α) implicitly defined by (7), it can be shown that limH→1 θS (1) = 1,

whereas previously we noted that limH→1 θS (0) = 0. Under the conditions of Lemma6, there exists a strictly positive θu0 . Then, the first inequality of (22) would hold for

H sufficiently close to 1. Under the same conditions, the last inequality of (22) holds

too with θu1 < 1 but limH→1 θS (1) = 1. Altogether, the conditions of the Lemma

and that H be sufficiently close to 1 constitute one set of sufficient conditions for

multiplicity.

With multiple steady-state equilibria, the presence of flippers in the market can

be fickle, especially when the equilibrium the market happens to be in is unstable. In

general, where there are multiple equilibria, any seemingly unimportant shock can dis-

locate the market from one equilibrium and move it to another, causing catastrophic

changes in flippers’ market share, turnover, and sales. To follow such discrete changes

in the activities of flippers can be significant fluctuations in housing price, a subject

we shall address in Section 6. And then in Section 7, we will calibrate the model to

several observable characteristics of the U.S. housing market to assess the quantita-

tive importance of such a channel of volatility. But next, we should first study the

problem of optimal flipping in the model housing market.

4 Efficiency

The planner, who is subject to the same trading and financing frictions that agents in

the model housing market face, chooses the fraction of mismatched households using

21

the services of flippers to maximize the utility flows over time that households derive

from matched owner-occupied housing net of the rental expenditures incurred; i.e.,

maxα

½Z ∞

0

e−rt (nMυ − nRq) dt¾, (25)

subject to (1)-(4), the equations of motions for nM and nU given by the differences be-

tween the LHS and RHS of (5) and (6), respectively, and some given initial conditions

for the two state variables.17

Lemma 7 In the steady state, the planner chooses

α =

⎧⎨⎩ 1

αE0

E (θS (1)) ≥ 0E (θS (αE)) = 0

E (θS (0)) ≤ 0, (26)

where

E (θ) =∂η

∂θ−µr + δ + η (θ)− θ

∂η

∂θ

¶z. (27)

The function E (θ), which can be thought of as the planner’s incentives to have

mismatched households selling to flippers,18 starts off equal to positive infinity at

θ = 0, is everywhere decreasing in θ, and ends up equal to some negative value

as θ → ∞. Then, with ∂θS (α) /∂α > 0, one and only one of the three cases in

(26) applies for a given parameter configuration. In particular, the third line implies

that the necessary and sufficient condition for any flipping to be optimal is that

E (θS (0)) > 0, which holds for small θS (0) with E (θ) a decreasing function. Other

things equal, a small θS (0) will arise out of a large housing supply, with which there

will be a sluggish and illiquid market. Conversely, in a tight and liquid market with

a large θS (0) due to a small housing supply, there will be rapid turnover no matter

what, in which case there is little to gain from any increase in turnover to offset the

additional resources incurred in the provision of rental housing while flipping takes

place. In comparing households’ private incentives to sell to flippers as given by bD in(17) with the planner’s incentives as given by E in (27) in Proposition 1 below, we

find that such excessive flipping is indeed possible in equilibrium.

17The equations of motion for nR and nF do not constitute independent restrictions given (1)-(4)

and the equations of motions for nM and nU as the two equations can be shown to be implied by

the former set of equations. Likewise, the initial values for nR and nF are not free variables but are

restricted by the initial conditions for nM and nU and (1) and (2).18Fornally, E (θ) denotes the sign of the steady-state shadow value of nR in the planner’s opti-

mization. When it is positive, the objective function in (25) increases in value as the planner sends

one more mismatched household to rental housing by making the household sell to a flipper.

22

Proposition 1

a. Equilibrium is efficient if

βH = 1−θ

η (θ)

∂η

∂θ≡ βe,

βF =

µ1− θ

η (θ)

∂η

∂θ

¶θ − αrrFθ − α

≡ βeF

at the optimum θ pair.

b. bD (θ,α) > E (θ) for θ at which∂ bD∂θ

> 0

and∂η

∂θ

1

η (θ)≤ 2

∙µr

rF− 1¶(1 + z) zβH

¸1/2. (28)

In case rF = r, Proposition 1(a) says that βeF = βe and the Hosios (1990) condition

holds exactly. The congestion externalities and the effects of imperfect appropriability

in the present model just cancel out and efficiency is obtained when the buyer’s share

of the match surplus, whoever the buyer is matched with, is just equal to the elasticity

of the matching function with respect to the measure of buyers. For rF < r, βeF < βe

though. This means that in particular, if rF < r but βF = βH = βe, the price that

flippers offer will be too attractive to give rise to excessive incentives for mismatched

households to sell in the investment market.

The conditions in part (b) are general sufficient conditions for excessive incentives

for flipping. If equilibrium is at where ∂ bD/∂θ > 0, households sell to flippers primarilyfor the high price in the investment market. The RHS of (28) is a positive real number

for rF < r, which will exceed the LHS of the condition for large θ given the concavity

of η (θ). The Proposition can then be interpreted to say that in a tight and liquid

market, possibly arising from a small housing supply, there is little efficiency gain

from flipping even though mismatched households may find the high price in the

investment market attractive, which in itself does not contribute to efficiency since

households would be paying a similarly high price to buy new houses later on when

they are selling at a high price in the investment market earlier.

Given that efficiency does not always increase with α and θ and that the market in-

centives to sell to flippers can be suboptimal or excessive, there is no reason to expect

that when there exist multiple equilibria, the more active equilibria are necessarily

more efficient. Most of all, the equilibria cannot be pareto ranked under all circum-

stances. Specifically, in a steady-state equilibrium where bD (θ,α) = 0, asset values

23

for matched and mismatched homeowners and renters are given by, respectively,19

VM =(r + βHη (θ)) υ

r (r + δ + βHη (θ)),

VU =βHη (θ) υ

r (r + δ + βHη (θ)),

VR =(βHrF − βF r) η (θ) υ

rF r (r + δ + βHη),

It is straightforward to verify that both VM and VU are increasing in θ. Any home-

owners — matched or mismatched — benefit from the higher housing prices in a tighter

market. But the asset value for households in rental housing VR is decreasing in θ

if βHrF < βF r, which is a necessary condition for the multiplicity of equilibrium

(equation (18) and Figure 3). In this case, would-be buyers are made worse off by the

higher housing prices in the tighter market. In the comparison between two steady-

state equilibria both at where bD (θ,α) = 0, homeowners are better off whereas rentersare worse off in the larger θ equilibrium than in the smaller θ equilibrium. Any two

such equilibria cannot then be pareto ranked. The same conclusion can be shown

to carry over to comparisons between a bD (θ,α) > 0 equilibrium and a bD (θ,α) = 0equilibrium and between a bD (θ,α) = 0 equilibrium and a bD (θ,α) < 0 equilibrium.5 Empirical implications

In this section, we shall explore several empirical implications of the model. To this

end, we begin with characterizing how the model housing market’s vacancy rate,

trading volume, and the turnover of houses and households vary with α.

5.1 Vacancy, trading volume, TOM, and TBM

In the model housing market, the entire stock of vacant house is comprised of houses

held by flippers. With a given housing stock, the vacancy rate is simply equal to

nF/H. A direct corollary of Lemma 1(b) is that:

Lemma 8 In the steady state, the vacancy rate for owner-occupied houses is increas-

ing in α.

Housing market transactions per time unit in the model are comprised of (i) αδnMhouses households sell to flippers, (ii) η (θ)nF houses flippers sell to households, and

19The asset values and housing prices referred to hereinafter are special cases of those in Lemma

12 in the Appendix. In particular, the equations for VM and VU are from (45) and (46), respectively,

whereas the equation for VR is from (44), evaluated at D (θ,α) = 0.

24

(iii) η (θ)nU houses sold by one household to another, adding up to an aggregate

transaction volume,

TV = αδnM + η (θ)nF + η (θ)nU . (29)

Lemma 9 In the steady state, TV is increasing in α.

Given that houses sold in the investment market are on the market for a vanish-

ingly small time interval and houses sold in the search market for a length of time

equal to 1/η (θ) on average, we may define the model’s average TOM as

αδnM

TV× 0 + η (θ) (nF + nU)

TV× 1

η (θ). (30)

Lemma 10 In the steady state, on average, TOM is decreasing in α.

TOM is a measure of the turnover of houses for sale. A more household-centric

measure of turnover is the length of time a household (rather than a house) has to

stay unmatched. We define what we call Time-Between-Matches (TBM) as the sum

of two spells: (1) the time it takes for a household to sell the old house, and (2) the

time it takes to find a new match afterwards. While the first spell (TOM) on average

is shorter with an increase in α, the second is longer as the increase in θ to accompany

the increase in α causes the buyer’s matching rate to fall. The old house is sold more

quickly. But it also takes longer on average to find a new match in a market with

more buyers and fewer sellers. To examine which effect dominates, write the model’s

average TBM as

α1

μ (θ)+ (1− α)

µ1

η (θ)+

1

μ (θ)

¶. (31)

where 1/μ is the average TBM for households who sell in the investment market20

and 1/η + 1/μ for households who sell in the search market.21

Lemma 11 In the steady state, on average, TBM is decreasing in α.

Lemma 11 may be taken as the dual of Lemma 1(b) (∂nM/∂α > 0). When

matched households are more numerous in the steady state, on average, they must

be spending less time between matches.

20The household sells the old house instantaneously. Given a buyer’s matching rate μ, the average

TBM is then 1/μ.21Let t1 denote the time it takes the household to sell the old house in the search market and

t2 − t1 the time it takes the household to find a new match after the old house is sold. Then thehousehold’s TBM is just t2. On average, E [t2] =

R∞0

ηe−ηt1³R∞

t1t2μe

−μ(t2−t1)dt2´dt1 = 1/η+1/μ.

25

5.2 Housing prices

No-intermediation equilibrium In the no-intermediation equilibrium, all hous-

ing market transactions are between pairs of end-user households at price22

pH =βH (η (θ) + r)− (1− βH)μ (θ)

(r + δ + βHη (θ)) rυ +

q

r, (32)

evaluated at θ = θS (0).

Fully-intermediated equilibrium In the fully-intermediated equilibrium, all houses

are first sold from mismatched households to flippers at price

pFB =βFη (θ) (υ + q)

(rF + βFη (θ)) r + (δ + (1− βF )μ (θ)) rF, (33)

in the investment market and then at price

pFS =βF (η (θ) + rF ) (υ + q)

(rF + βFη (θ)) r + (δ + (1− βF )μ (θ)) rF, (34)

from flippers to end-user households in the search market, both evaluated at θ =

θS (1). With houses sold by households to flippers on the market for a vanishingly

small time interval and houses sold by flippers to households for, on average, 1/η (θ) >

0 units of time, prices and TOM in the model housing market, as in the real-world

housing market, are positively correlated in the cross section, given that by (33) and

(34), pFB < pFS. Besides, with pFB < pFS, the model trivially predicts that houses

bought by flippers are at lower prices than are houses bought by non-flippers. Both

Depken et al. (2009) and Bayer et al. (2011) find evidences of such flipper-buy

discounts in their respective hedonic price regressions.

Partially-intermediated equilibrium In a steady-state equilibrium in whichmis-

matched households sell in both the investment and search markets, in addition to

the two prices

pFB =βFη (θ)

rF (r + δ + βHη (θ))υ, (35)

pFS =βFη (θ) + βF rF

rF (r + δ + βHη (θ))υ, (36)

for transactions between a flipper and an end-user household, there will also be trans-

actions between two end-user households, carried out at price

pH =βFη (θ) + βHrF

rF (r + δ + βHη (θ))υ. (37)

22Equation (32) is from (41) evaluated at α = 0; (33) and (34) are from (49) and (48), respectively,

evaluated at α = 1; (35), (36), and (37) are from (43), (42), and (41), respectively, all evaluated at

D (θ,α) = 0.

26

For βF > βH , pFB < pH < pFS. Just as in the fully-intermediated equilibrium,

a positive relation between prices and TOM holds in the cross section and houses

bought by flippers are at lower prices. Moreover, here houses sold by flippers are sold

at a premium over houses sold by one end-user household to another. Such flipper-sell

premiums are also found to exist in Depken et al. (2009) and Bayer et al. (2011).

Across equilibria Across steady-state equilibria, θ is largest in the equilibrium

where flippers are most numerous. Then, prices should be highest in the given equi-

librium where the competition among buyers is most intense.

Proposition 2 Across steady-state equilibria in case there exist multiple equilibria,

housing prices in both the search and investment markets are highest in the equilibrium

with the tightest market and lowest in the equilibrium with the most sluggish market.

5.3 Correlations among prices, TV, vacancy, TOM, and TBM

Now, a direct corollary of Proposition 2 and Lemmas 8-11 is that:

Proposition 3 Across steady-state equilibria in case there exist multiple equilibria,

prices, TV, and vacancies increase or decrease together from one to another equilib-

rium, whereas the average TOM and TBM move with the former set of variables in

the opposite direction.

Interest rate shocks In a typical asset pricing model, the price of an asset falls

when the interest rate goes up. The same tends to hold in the present model. Specif-

ically, in the no-intermediation equilibrium, an increase in r, by (32), leads to a lower

pH for sufficiently large θS (0) and/or q. Similarly, in the fully-intermediated equi-

librium, by (33) and (34), respectively, both pFB and pFS are decreasing in rF . But

in either equilibrium with α remaining fixed at 0 or 1, market tightness, vacancies,

turnover, and sales are all invariant to the respective interest rate shocks.

In a partially-intermediated equilibrium, prices in the search market, pFS and

pH , as well as in the investment market pFB, are decreasing in rF , just as they are

in the fully-intermediated equilibrium. Housing prices in a partially-intermediated

equilibrium, however, can also vary to follow any movements in θ triggered by the

given interest rate shock — when the market becomes tighter in particular, prices are

also higher. Hence, if a given positive (negative) shock to rF should cause α and

therefore θ to decrease (increase), there will be lower (higher) housing prices to follow

because of a direct negative (positive) effect and of an indirect effect due to the exit

(entry) of flippers. When the two effects work in the same direction, the interest rate

shock can cause significantly more housing price volatility than in a model that only

allows for the usual effect of interest rates on asset prices.

27

A positive shock to rF need not cause α and θ to fall though. In case there exist

multiple equilibria, the shock can possibly dislocate the market from a given equilib-

rium and send it to another equilibrium. In case the direct effect of an interest rate

shock and the indirect effect via the movements in θ affect housing prices differently,

in what direction housing prices will move cannot be unambiguously read off from

(35)-(37). To proceed, we solve bD (θ,α) = 0 for rF and substitute the result into

(35)-(37), respectively,

pFB =βHη (θ) + ((βF − βH)αS (θ)− (1− βH))μ (θ)

r (r + δ + βHη (θ))υ +

q

r, (38)

pFS =βHη (θ) + βF r + ((βF − βH)αS (θ)− (1− βH))μ (θ)

r (r + δ + βHη (θ))υ +

q

r, (39)

pH =βHη (θ) + βHr + ((βF − βH)αS (θ)− (1− βH))μ (θ)

r (r + δ + βHη (θ))υ +

q

r. (40)

The three expressions are independent of rF — whatever effects a given change in

rF will have on housing prices are subsumed through the effects of the change in θ

that follows the change in rF obtained from holding bD (θ,α) = 0. To evaluate the

effects of rF on housing prices is to simply check how these three expressions behave

as functions of θ.

Proposition 4 Across steady-state equilibria and holding bD (θ,α) = 0, a shock torF , whether positive or negative, will cause housing prices to increase (decrease), as

long as to follow the interest rate shock are increases (decreases) in α and θ.

By Proposition 4, the indirect effect of an interest rate shock on housing prices

through the entry and exit of flippers and then in market tightness always dominates

the direct effect shall the two be of opposite directions. A surprising implication is

that housing prices can actually go up in response to an increase in flippers’ cost of

financing, if to follow the higher interest rate is also a heightened presence of flippers

in the market. In any case, a direct corollary of Lemmas 8-11 and Proposition 4 is

that:

Proposition 5 Across steady-state equilibria and holding bD (θ,α) = 0, a shock to rFwill cause housing prices, TV, and vacancies to move in the same direction, whereas

the average TOM and TBM will move in the opposite direction.

In the above, we have restricted attention to analyzing how changes in rF alone

may affect housing prices. It turns out that many of the implications continue to

hold for equiproportionate increases or decreases in r and rF . Proposition 7 in the

Appendix contains the details.

28

6 Time-series Relations among Housing price, TV,

and Vacancy

By Propositions 3, 5, and 7 in the Appendix, any movement from one to another

steady-state equilibrium would involve housing prices, TV, and vacancies all going

up or down together. The positive time-series relation between housing prices and

the volume of transaction is well known and numerous models have been constructed

to account for it. In Kranier (2001), for instance, a positive but temporary preference

shock can give rise to higher prices and a greater volume of transaction, whereas Diaz

and Jerez’s (2013) analysis implies that an adverse shock to construction will shorten

TOM, andmay possibly lead to higher prices and a greater volume of transaction. The

paper by Ngai and Tenreyro (2014) studies the comovement in prices and sales over

the seasonal cycle and they argue that increasing returns in the matching technology

play a key role in generating such cycles.

Unique to our analysis is that vacancies should also move in the same direction

with prices and TV. In contrast, in both the Kranier and the Diaz and Jerez’s models,

the increase in sales should be accompanied by a decline in vacancy — given that when

a house is sold, it is sold to an end-user, who will immediately occupy it, vacancies

must decline, or at least remain unchanged. In Ngai and Tenreyro’s model, households

are assumed to move out of their old houses and into rental housing immediately after

they become mismatched. Then, any and all houses on the market are vacant houses

and given the assumed increasing-returns-to-scale matching technology, vacancies rise

and fall with prices and the volume of transaction in the seasonal cycle. Amismatched

household in their model, however, could well have stayed in the old house and avoided

rental housing and the payment thereof until it has successfully sold the old house.

In this alternative setup, the stock of vacant houses only includes houses held by

people who have bought new houses before they manage to sell their old ones. Then,

it is no longer clear that vacancies must rise and fall with prices and the volume of

transaction in the seasonal cycle of Ngai and Tenreyro.

Figure 7 depicts the familiar positive housing price-transaction volume correlation

for the U.S. for the 1981Q1 to 2011Q3 time period.23 The usual housing market search

model predicts that vacancies should decline in the housing market boom in the late

1990s to the mid 2000s and rise thereafter when the market collapses around 2007.

Figures 8 and 9 show that any decline in vacancy is not apparent in the boom.24 In

23Housing Price is defined as the nominal house price, which is the transaction-based house price

index from OFHEO (http://www.fhfa.gov), divided by the CPI, from the Federal Reserve Bank at

St. Louis. We set Housing Price at 1981Q1 equal to 100. Transaction is measured by the quarterly

sales in single-family homes, apartment condos, and co-ops, normed by the stock of such units. The

sales data are from the Real Estate Outlook by the National Assoication of Realtors, complied by

Moody’s Analytics. The housing stock is defined as the sum of owner-occupied units and vacant

and for-sale-only units. The data are from the Bureau of Census’s CPS/HVS Series H-111 available

at http://www.census.gov/housing/hvs/data/histtabs.html.24Vacancy rate is obtained by dividing the number of vacant and for-sale-only housing units by

29

Figure 7: Price and Transactions

fact, if there is any comovement between vacancies on the one hand and prices and

sales on the other hand in the run-up to the peak of the housing market boom in 2006,

vacancies appear to have risen along with prices and the volume of transaction. In

a literal interpretation of our model, vacancies should fall very significantly to follow

the market collapse since 2007. Apparently, the decline in vacancy in Figures 8 and 9

in the post-2007 period is modest, compared to the increase in the boom years. Two

forces absent in our analysis — the massive amount of bank foreclosures and unsold

new constructions in the market bust — may have accounted for the slow decline in

vacancy since 2007.

In a more systematic analysis, we first verify that in the 1981Q1 to 2011Q3 sample

period, all three variables are I(1) at conventional significance levels. Next, we test

for cointegration. Assuming the absence of any time trends and intercepts in the

cointegrating equations, both the Trace test and the Max-eigenvalue test indicate

two such equations, whose normalized forms read

Price — 6045.51×Vacancy = 0,Transaction — 0.74×Vacancy = 0,

which together imply that the three variables do tend to move in the same direction

from one to another long-run equilibrium over time.25

the housing stock as defined in the prevous note.25With other time trend and intercept assumptions, either one or both of the tests suggest that

30

Figure 8: Price and Vacancy

Figure 9: Transactions and Vacancy

31

7 Quantitative predictions on volatility

Given the possible multiplicity of equilibrium and that an interest rate shock may have

important effects on the extent of intermediation, the model can be consistent with

a volatile housing market. The question remains as to how important quantitatively

such channels of volatility can be. In this section, we calibrate the model to several

observable characteristics of the U.S. housing market and study by how much housing

prices can fluctuate across steady-state equilibria and in response to interest rate

shocks.

To begin, we take a time unit in the model to be a quarter of a year and assume

a Cobb-Douglas matching function with which η (θ) = aθb. We set a priori the

mismatch rate δ = 0.014 to calibrate a two-year mobility rate of 11.4% for owner-

occupiers reported in Ferreira et al. (2010) and b = 0.84, which is the elasticity

of the seller’s matching hazard with respect to the buyer-seller ratio reported in

Genesove and Han (2012). Next, the parameters a andH and the share of mismatched

households selling to flippers α are chosen to calibrate:

1. a quarterly transaction rate of owner-occupied houses of 1.78%

2. a vacancy rate of owner-occupied houses of 1.84%

3. the share of houses bought by flippers among all transactions of owner-occupied

houses equal to 19%

The first two targets are, respectively, the average quarterly transaction rate and the

average vacancy rate for the period 2000Q1-2006Q4, calculated from our dataset for

the plots in Figures 7-9 and the estimations in Section 6. Estimates of the share

of houses bought by flippers come from two sources: the 25% investors’ share of all

new purchase mortgages in the whole of the U.S. in Haughwout et al. (2011) in

the 2000Q1-2006Q4 time period and the 13.7% housing market transactions share

for houses sold again within the first two years of purchase in the metropolitan Las

Vegas area in Depken et al. (2009) in the same time period. Because an investor in

Haughwout et al. (2011) may intend to hold the house as a long-term investment,

the 25% share is probably an overestimate of the true flippers’ share. Because not all

houses bought for short-term flips can actually be sold within two years, the 13.7%

there exist only one or as many as three cointegrating equations. In a single cointegrating equation

with non-zero coefficients for all three variables, at least two of the three coefficients must be of

the same sign. Then, the two variables concerned must move in opposite directions across long-run

equilibria. With as many cointegrating equations as the number of variables, there exist definite

long-run values for the three variables, which rules out the possibility of the system moving from

one to another long-run equilibrium altogether. Restricting a priori to two cointegrating equations

in the estimation, however, we always obtain two equations whose coefficients have the same signs

as those in the system above whatever the trend and intercept assumptions are. Then, any long-run

movements of the three variables must be in the same direction.

32

share in Depken et al. (2009) is probably an underestimate of the true flippers’s

share. Our 19% target is obtained by taking a simple average of the two estimates.

Given the targets, denoted as xi, i = 1, 2, 3, respectively, we choose a, H, and α

to

min

(3Xi=1

µxi − bxixi

¶2),

where the bxi’s are the model’s calibrated values of the corresponding targets26, yield-ing a = 0.085, H = 0.865, and α = 0.25 at which the calibrated values of the three

targets are reported in the second column of Table 1.

Table 1: Calibration Targets and Calibrated Model ValuesTargeted value Calibrated value

Transaction rate (quarterly) 0.0178 0.016

Vacancy rate 0.0184 0.018

Flippers’ share in transactions 0.19 0.19

Thus far in the calibration, we have effectively identified αS = 0.25 as equi-

librium. For equilibrium to be indeed at α = 25, we need to pick the values for

{q, υ,βH ,βF , r, rF} to force αO = 0.25 as well. Since only the ratio z = q/υ, but notthe levels of the two parameters, matters for the value of αO and the comparison of

prices, we first normalize υ = 1. We then obtain an estimate of z (or equivalently

q) equal to 1.43 from the results in Anenberg and Bayer (2013). The details are in

Appendix 10.2. Next, we set r = 0.02 for an annual rate of 8% to match the usual

30-year fixed-rate mortgage rate. Lastly, for the lack of any obvious empirical coun-

terpart, we set the household-seller’s bargaining strength βH = 0.5. Then, for each

of βF = 0.5, 0.6, 0.65, 0.7, and 0.8, we look for the value of rF at which αO = 0.25.

The results are shown in Table 2.27,28

26The bxs are equal to TV/H, nF /H, and αδnM/TV for the model’s transaction rate, vacancy

rate, and the flippers’ transaction share, respectively. The minimization is carried out via a grid

search with a grid size of 0.005 for each of a, H, and α, subject to a ≤ 1.2 and H ≥ 0.6. The firstconstraint is for expediency in the grid search and is not binding. Given that H in the model is the

stock of owner-occupied houses relative to the population of households demanding such housing,

anything near the bound of the second constraint is probably unreasonable.27A rF below r by a few percentage points can make sense if flippers, but not end-user households,

tend to be all-cash investors. Herbert et al. (2013) report that the majority of investors acquiring

foreclosures are indeed all-cash buyers. Even though no comparable evidence is available for other

properties, it would not be surprising that cash is often used too. Moreover, investors may also make

use of mortgages with zero initial or negative amortization, short interest rate reset periods, or low

introductory teaser interest rates. Such mortgages obviously are ideal for flippers who plan to sell

quickly for short-term gains. Amromin et al. (2012) find that borrowers who take out such “complex”

mortages are usually high income individuals with good credit scores. Foote et al. (2012) find that

periods of interest rate resets do not tend to trigger significant increases in defaults, consistent with

the finding of Amromin et al. that the borrowers of such mortgages are sophisticated investors.28Notice that the model does not require rF < r for flippers to survive or for the multiplicity of

33

For the last two pairs of βF and rF in Table 2, the α = 0.25 equilibrium is the

unique equilibrium. For the first three pairs, there are two other equilibria each beside

the α = 0.25 equilibrium. Table 3 reports the prices in these equilibria. For instance,

for βF = 0.5 and rF = 3.1% per annum, the three equilibria are at α = 0, 0.25, and

1, respectively.29 The price p, on the next row, is the average of pH , pFB, and pFS,

weighted by the shares of transactions taking place at the respective prices, with p

in the smallest-α equilibrium set equal to 1. Evidently, the volatility arising from

the multiplicity is non-trivial, with average prices differing by up to 23% across the

equilibria.

Table 2: Calibrated βF and rF for αO = 0.25

βF 0.5 0.6 0.65 0.7 0.8

rF 0.0077 0.0091 0.0099 0.0106 0.012

rF (annual basis) 3.1% 3.7% 4% 4.23% 4.8%

Table 3: Multiple EquilibriaβF 0.5 0.6 0.65

rF (annual basis) 3.1% 3.7% 4%

α 0 0.25 1 0 0.25 1 0.25 0.71 1

p 1 1.1 1.2 1 1.11 1.23 1 1.08 1.11

Table 4: Housing Prices and Interest Rates, βF = 0.7

rF (annual basis) 3.5% 4.21% 4.22% 4.23% 4.27% 6%

α = 0 0.89 0.89

α = 0.25 1

α = 0.63 1.07

α = 1 1.13 1.12

To study the response of housing prices to interest rate shocks, we report in Table

4 average housing prices p for various small deviations of rF from a benchmark of

rF = 4.23% and βF = 0.7 at which equilibrium is unique at the calibrated value

of α = 0.25. Fixing βF = 0.7, for all values of rF under consideration, equilibrium

remains unique. The entries in the table are normed by the average equilibrium price

at the benchmark rF . Here, housing prices hardly move to follow a given interest rate

equilibria. For smaller values for z, we can force αO to be equal to 0.25 for much larger rF .29At α = 1, in the steady state, one half of all sales are purchases made by flippers. This is just

about equal to the peak investor share in the “bubble states” reported in Haughwout et al. (2011).

34

shock if the shock has not caused any changes in equilibrium α. But when the given

interest rate shock does cause α to change significantly, it also leads to significant

changes in housing prices. Specifically, a decline in rF from 6% per annum to 4.27%

per annum causes no noticeable change in p when the given movement in rF has no

effect on α. But a further decline in rF from 4.27% per annum to 4.21% per annum

now causes p to increase by 26% as α rises from 0 to 1 in the meantime. Thereafter,

p remains essentially unchanged from any additional decline in rF as α has already

reached the upper bound of 1. All this suggests that the response of housing prices to

interest rate shocks can appear erratic and unpredictable. Before a given threshold

rF is reached, the response is at most moderate. When rF crosses the threshold to

trigger the entry of flippers, the housing market can become significantly tighter and

housing prices significantly higher as a result.

8 Extensions

The model we studied in this paper is clearly a very special model. In Leung and

Tse (2016a), we study how the major results of the paper may be affected under

competitive search, a two-house-limit liquidity constraint for households, allowing for

investors choosing between short-term flips and long-term investments and letting

mismatched households sell to housing market intermediaries right before buying a

new house. Below are the summaries of the findings.

8.1 Competitive search

In our model, the multiplicity of equilibrium arises out of flipper-sellers passing onto

household-sellers their advantages in bargaining to the fullest extent possible in a tight

and liquid market. A natural question to ask is whether the multiplicity is special to

price determination by bargaining as we have assumed or whether similar conclusions

hold under the alternative assumption of price determination via competitive search.

In the competitive search version of our model, based on Mortensen and Wright

(2002), the search market is segmented into submarkets, each of which is controlled

by a competitive market maker, who charges entry fees for other agents for buying

and selling in his submarket in return for regulating the transaction price and market

tightness at some pre-specified levels. The household-buyers, household-sellers and

flipper-sellers choose which submarket — among the options available — to enter into

to maximize the respective expected returns of buying and selling. In this setting,

we find that in equilibrium,

α =

⎧⎨⎩ 1

αC0

C (θS (1)) ≥ 0C (θS (αC)) = 0

C (θS (0)) ≤ 0,

35

where

C (θ) =∂η

∂θ+

µη (θ)− θ

∂η

∂θ

¶µr

rF− (1 + z)

¶− z (δ + r) ,

can be thought of as the incentives of mismatched households to sell in the invest-

ment market under competitive search — the counterpart to bD (θ,α) in (17) underbargaining and E (θ) in (27) for efficiency.

First notice that if rF = r , C (θ) = E (θ) and hence, as is well known, competitive

search is efficient. For rF 6= r, however, C (θ) 6= E (θ). Most importantly for our

purpose, for

rF <r

1 + z,

C (θ), like bD (θ,α) for rF < r, is U-shaped, first decreasing but eventually increas-ing. This of course opens up the possibility of multiplicity. Indeed, this necessary

condition for multiplicity is the same necessary condition for multiplicity in Lemma

5 and Figure 3 for βF = βH . Thus, as long as flippers possess a sufficiently large fi-

nancing advantage, they can attract mismatched households to sell in the investment

market when the search market is particularly tight, in addition to when the search

market is particularly sluggish, whether prices in the search market are determined

by bargaining or via competitive search.

8.2 Two-house-limit liquidity constraint

In some version of the housing market search model, most notably in Wheaton (1990),

there is not any rental housing at all but rather a mismatched household must stay

in its old house while searching for a new match, and then the old house will be

put up for sale only after the household has found a new house to move into. In

this environment, a household will be holding as many as two mismatched houses if

the household is hit by a moving shock again before it is able to sell the previously

mismatched house. If there is a two-house-limit liquidity constraint, the household

will then be prevented from entering the search market as a buyer until it is able to

sell one of the two mismatched houses that it is now holding. A liquidity provision

role for flippers arises. Indeed, a household may wish to sell to flippers right after it

is moving to a new house from the old mismatched house. By doing so, the household

will never be holding two houses at any moment in time. In all, we find that when

more households sell to flippers, either right after finding new houses to move into or

right after being hit by two successive moving shocks, there will be a tighter search

market.

In this setup, flippers, like those in the present model, may be able to lure house-

holds to sell in the investment market not just when the search market is sluggish but

also when it is tight if they possess a large enough financing/bargaining advantage. In

this model though, there is not any relationship between the activities of flippers and

36

the vacancy rate since the latter is simply equal to the difference between the hous-

ing stock and the population of households, in the entire absence of rental housing.

Furthermore, unlike the present model, for efficiency, all households should use the

services of flippers since nobody would be incurring any rental expenditures during

which flipping takes place. And in case there exist multiple equilibria, a more active

equilibrium should pareto dominate a less active one, with any household owning at

least one house at any moment.

8.3 Short-term flips versus long-term investment

In reality, there can be two strategies for housing market investments — short-term

flip versus long-term investment in which an investor holds the house for an extended

period of time, earning the rental revenue in the interim and in anticipation for a

certain capital gain in the medium to long term. In a summary of case studies of four

metropolitan areas in the U.S., Herbert et al. (2013) report that both investment

strategies were commonly adopted by housing market investors in the wake of the

collapse of the housing market in the U.S. in 2007. In the early phase when prices

appeared to have reached the lowest level, most investments were found to be short-

term flips, whereas in the latter phase when the market appeared to have stabilized,

most investments were found to be medium- to long-term investments.

In the present model, the supply of rental housing is assumed perfectly elastic at

some exogenously given rental. We could have chosen to assume the arguably more

realistic setting in which the stock is exogenously given while the market rental is

determined in equilibrium. All the same, underlying either setting is the presumption

that the rental and owner-occupied housing stocks are two separate stocks. By all

means, a richer analysis would allow for the same housing stock to serve as both

rental and owner-occupied housing. In this revised model, just as in the present

model, mismatched households choose between selling in the investment market or

offering their houses for sale in the end-user search market. Unlike the present model,

the specialist investors in the revised model may choose to offer their properties for

sale and/or for rent. It turns out that in this setup, the only kind of steady state is one

in which at least a fraction of mismatched households choose to sell to investors while

investors choose to offer their properties both for sale and for rent. Any such steady

state, however, can be equilibrium only under selected values of the housing stock

and investors’ cost of financing, just as a fully- or partially-intermediated equilibrium

exists only for some subset of the parameter space in the present model. When

the conditions are not met, no steady-state equilibrium exists in the revised model.

This result is perhaps highly suggestive for in reality, housing market investors do

predominantly choose whether to flip or to invest long term during different phases

of the housing price cycle as reported in Herbert et al. (2013). No matter, to analyze

a model that allows for investors choosing between the two investment strategies, it

becomes imperative to study the full dynamics. Undoubtedly, this can be a very

37

fruitful exercise towards a fuller understanding of the dynamics of housing market

investment but is best to be left for future research.30

8.4 Selling to housing market intermediaries right before

buying a new house

We have in this paper assumed that a mismatched household must either sell to a

flipper or to an end-user household before it can start looking for a new house. Strictly

speaking, given the assumption of instantaneous sale in the investment market, the

household can choose to stay in the old house while searching for a new one and then

sell the old house to a flipper only right before the household buys the new house.31

This means that, a one-house-limit liquidity constraint notwithstanding, mismatched

households should be able to enter the search market as buyers before selling their

old houses.

Given that all mismatched houses are for sale in the search market, whoever

their owners are, had we allowed for any and all mismatched households to enter the

search market as buyers right away, the tightness in the search market would not

depend in any way on how many mismatched households choose to sell to flippers

in the first instance, if any mismatched households choose to do that at all. In this

alternative setting, market tightness depends solely on the housing stock, among

other factors, and is completely isomorphic to price determination in equilibrium.

Moreover, flippers’ liquidity provision role is wholly fulfilled so long as they are in the

market ready to buy up any houses would-be buyers need to sell just before buying.

Then, unless flippers possess some large enough financing/bargaining advantage, α

must be just equal to zero in equilibrium. In sum, equilibrium is guaranteed unique

and any sale to flippers that takes place at the moment households first becoming

mismatched must arise out of flippers’ financing/bargaining advantage.

True, with a frictionless investment market in place, our assumption that mis-

matched households must rid themselves of their old houses first before entering the

search market as buyers is ad hoc. We could have included some additional techni-

cal details to better justify the assumption.32 But we think it is more constructive

to simply note that the assumption is a well-motivated assumption. In reality, the

investment market for the housing asset is by no means completely free of frictions.

30Indeed, Head et al. (2014) have explored a number of interesting implications of a model that

allows households and real estate developers a choice between the two strategies. They do not allow

for specialist investors in their model like we do though.31The households who move within a given city in Head and Lloyd-Ellis (2012) can do just that.

There, households do not face any liquidity constraints and the assumption of having housing market

intermediaries is merely a simplifying assumption to facilitate analysis.32For example, consider a discrete time version of the model. Say a period is divided into two

subperiods where the investment market is open in the first subperiod only and the search market

is open next in the second subperiod. In Leung and Tse (2016a), we propose three other alternative

justifications.

38

Sales in the market are certainly not instantaneous. If a household is not able to

sell the old house quickly enough to pay off the mortgage for the house, it can face

considerable difficulties in getting a mortgage for the new house. A more realistic and

arguably more rigorous analysis is to model the investment market as a search market

too. A frictionless investment market is best thought of as a simplifying assumption

or, as we remarked in Section 2, as the limit of a frictional market when the cost of

entry for investors tends to zero.

9 Concluding remarks

Housing market flippers can be arbitraging middlemen like those we model in this

paper, intermediaries who survive on the basis of superior information, or momentum

traders blindly chasing the market trend. Our analysis suggests a number of empirical

implications to distinguish between these theories of flipping. First and foremost, if

flippers are predominantly momentum traders instead of specialist middlemen, there

should not be any significant flipper-buy discounts and flipper-sell premiums. But

such discounts and premiums do exist and they are sizeable, as reported in Depken

et al. (2009) and Bayer et al. (2011). Second, in a housing market boom fed by

the entry of momentum traders, there can and usually will be sales and purchases

among flippers. While in a more elaborate theory of intermediation as in Wright and

Wong (2014), this can also happen. But this does not seem like a robust implication

of a theory of middlemen in the housing market. Third, a theory of housing price

speculation should imply that prices should stay at the peak for at most a short

while and then fall right afterwards. In our model, the market can conceivably move

from a low-price to a high-price steady-state equilibrium and then just stays at the

new equilibrium for any length of time. Relatedly, speculators should only buy in

an up market, whereas in our model, there can be multiple locally stable steady-

state equilibria involving flipping. Indeed, in our model, the gross returns to flipping,

pFS/pFB, by (11), is actually higher in a lower-price less active equilibrium than in a

higher-price more active equilibrium. Lastly, if housing market middlemen are mostly

“market edge” investors who survive on the basis of informational advantage rather

than investors who have more flexible and less costly financing like those in our model,

they should use ordinary mortgages as much as end-user households do. Herbert et

al. (2013) report that there was little evidence of bank lending to investors acquiring

foreclosed properties in the wake of the 2007 housing market meltdown in the U.S.

Instead, the great majority of acquisitions were bought with cash. In addition, in

the housing market boom in the U.S. before 2007, there is ample evidence, as we

remarked in note 27, that investors took advantage of “complex mortgages” more

than ordinary households did. Other than observations on the choices of financing,

the two theories may also be distinguished by what market conditions under which

flipping takes place. In our model, flipping tends to occur in particularly tight as

well as particularly sluggish markets — a prediction that is hard to envisage to come

39

from a theory of housing market intermediaries who survive on the basis of superior

information.

In the U.S., house flipping is thought to often involve renovating before selling

rather than simply buying and then putting up the house for sale right after. In this

line of thinking, the returns to flipping are more about the returns to the renova-

tions investment than the returns to holding the houses on behalf of the liquidity-

constrained owners. The question then is what prevents the original owners them-

selves from earning the returns on the investment. A not implausible explanation is

that many original owners lack the access to capital to undertake the investment, just

as the original owners in our model lack the access to capital to hold more than one

house at a time. Thus, at a deeper level, our model is not just a model of buy-and-sell

flips but should also encompass, with suitable modifications, buy-renovate-sell flips.

40

10 Appendix

10.1 Lemma and Proposition

Lemma 12 If max {VR + pFB, VU} = VU ,pH = {(βH (η + r) (rF + βFη)− (1− βH) ((1− α)βFη + (1− αβF ) rF )μ) υ

+(r + δ + βHη) (rF + βFη) q} /GU , (41)

pFS = βF (η + rF )(r + βHη − μ (1− α) (1− βH)) υ + (r + δ + βHη) q

GU, (42)

pFB = βFη(r + ηβH − μ (1− α) (1− βH)) υ + (r + δ + ηβH) q

GU, (43)

VR = {((1− α) (1− βH)βFηrH + (1− βF ) ηβHαrF + (1− βH − α (βF − βH))

×rrF )μυ − (r + δ + βHη) (rF + βFη) rq} / (rGU) , (44)

VM =(r + βHη) υ

r (r + δ + βHη), (45)

VU =ηβHυ

r (r + δ + βHη), (46)

where

GU = (r + δ + ηβH) ((rF + ηβF ) r + (1− βF ) rFμα) .

If max {VR + pFB, VU} = VR + pFB,pH = {(βH (η + r) (rF + βFη)− μ (1− βH) (rF + (1− α)βFη − αβF rF )) υ

+((1− βH) δrF + (r + βHη) (rF + βFη)) q} /GF , (47)

pFS =βF (η + rF ) ((r + βHη − (1− βH) (1− α)μ) υ + (r + βHη) q)

GF, (48)

pFB =βFη ((βHη + r − μ (1− α) (1− βH)) υ + (βHη + r) q)

GF, (49)

VR = {(((1− α)βFη + rF ) (1− βH) r + (1− βF )βHηαrF − (βF − βH)αrF r)

×μυ − (r + ηβH) (rF r + δrF + βFηrH) q} / (rGF ) , (50)

VM =(r + βHη) (((rF + βFη) r + (1− βF ) rFμα) υ − rF δq)

rGF, (51)

VU =ηβH (((rF + βFη) r + (1− βF ) rFμα) υ − rF δq)

rGF, (52)

where

GF = (r + βHη) (rF r + δrF + βFηr + (1− βF ) rFμα)− (1− βH) (1− α)μδrF .

41

Lemma 13

a. Write

SRF = VM − VR − pFB,SRU = VM − VU ,

respectively, as the match surpluses in matches between a buyer in rental hous-

ing and a flipper-seller and a mismatched household-seller. If D (θ,α) ≥ 0,

SRF ≥ 0. If D (θ,α) ≤ 0, SRU ≥ 0.b. max {VR + pFB, VU} ≥ 0.

Proposition 6 Equilibrium exists for all {r, rF , υ, q, δ,βF , βH , H} tuple.

Proposition 7

a. In the fully-intermediated equilibrium, equiproportionate increases in r and rFlower housing prices. The same effect is felt in the no-intermediation equilib-

rium for sufficiently large θS (0) and/or q.

b. In a partially-intermediated equilibrium,

i. equiproportionate increases in r and rF , holding θ fixed, lower housing

prices;

ii. across steady-state equilibria and holding bD (θ,α) = 0, equiproportionatechanges in r and rF , whether positive or otherwise, cause pFB to increase

(decrease) as long as to follow the interest rate shocks are increases (de-

creases) in θ and α for βF = βH = 1/2 and r/rF ∈ [0, 1 + z]; the sameeffect is felt on pFS and pH for r/rF in neighborhoods of r/rF = 0, 1, and

1 + z.

10.2 Calibrating z = q/υ

In the model housing market, the flow payoffs for matched owner-occupiers, mis-

matched owner-occupiers, and buyers in rental housing are equal to υ, 0, and −q,respectively. In this case then, the difference between the flow payoffs of matched

and mismatched owner-occupiers is equal to υ, and that between mismatched owner-

occupiers and buyers is q. Anenberg and Bayer (2013) report estimates on

1. mean flow payoff for matched owner-occupiers 0.0273

2. flow payoff for owner-occupiers mismatched with their old houses 0.024, com-

prising 30% of all mismatched households

42

3. flow payoff for owner-occupiers mismatched with the metro area 0.0014, com-

prising 70% of all mismatched households

while normalizing the flow payoff of buyers to 0. We may thus equate υ = 0.0273−0.024 = 0.00033 and q = 0.024 so that z = 8 for households who are mismatched with

their old houses and υ = 0.0273− 0.0014 = 0.0259 and q = 0.0014 so that z = 0.054for households who are mismatched with the metro area. Taking a weighted average

of the two estimates gives a value for z equal to 2.49. Alternatively, one may take the

weighted average first before taking the ratio: υ = 0.0273−0.03×0.024−0.7×0.0014and q = 0.03 × 0.024 + 0.7 × 0.0014, so that z = 0.43. Since it is not clear in the

context of our model which method is conceptually better than the other, we resort

to taking a simple average of 2.49 and 0.43 to obtain a value of 1.43 for z.

10.3 Proofs

Proof of Lemmas 1 and 2 Solve (1)-(6) for

nM =ηH

η + δ, (53)

nF =αδH

δ + η, (54)

nU =(1− α) δH

δ + η, (55)

nR =η (1−H)− (1− α) δH + δ

δ + η. (56)

Equation (7) is obtained by substituting (54)-(56) into

θ =nR

nU + nF.

The LHS of (7) is equal to (1− (1− α)H) δ > 0 at θ = 0 but is negative for arbitrarily

large θ given the concavity of η. A solution is guaranteed to exist. Differentiating

with respect to θ yields∂η

∂θ(1−H)−Hδ,

which is positive for small θ but negative otherwise. The solution to (7) must then

be unique, and that the LHS is decreasing in θ at where it vanishes. Given that the

LHS of the equation is increasing in α, ∂θS/∂α > 0. The lower and upper bounds

θS (0) and θS (1) are given by the respective solutions to

δ + η (1−H)− (1 + θ)Hδ = 0, (57)

δ + η (1−H)− θHδ = 0, (58)

43

both of which are easily seen to be positive and finite.

Solve (7) for

α =θδH − (1−H) (η + δ)

δH, (59)

and substitute into (54)-(56), respectively,

nF =θδH − (1−H) (η + δ)

δ + η, (60)

nU =η (1−H) + δ (1− θH)

η + δ, (61)

nR =θδH

η + δ. (62)

The comparative statics in the Lemma can be obtained by differentiating (53) and

(60)-(62), respectively, with respect to θ, and then noting that ∂θS/∂α > 0. The

boundary values for nF and nU are obtained for setting α = 0 and α = 1 in (54) and

(55), respectively.

Proof of Lemma 3 Substituting from (54) and (55) into nFnF+nU

yields the result

of the Lemma.

Proof of Lemmas 4 and 5 and the construction of Figure 3 Given that

limθ→0 η = 0 and limθ→∞ μ = 0,

limθ→0

bD = (1− βH − α (βF − βH))μ− (δ + r) z =∞,

limθ→∞

bD = µβF rrF − βH − zβH¶η − (δ + r) z. (63)

Differentiating,

∂ bD∂θ

=

µβF

r

rF− βH − zβH

¶∂η

∂θ+ (1− βH − α (βF − βH))

∂μ

∂θ. (64)

For rF ≥ r, the expressions in both (63) and (64) are negative for sure. In this case,as a function of θ, bD starts out equal to positive infinity and falls continuously belowzero. Then, there exist some θd1 and θd0 that satisfy

bD ¡θd1, 1¢ = 0 and bD ¡θd0, 0¢ = 0,respectively. Given that bD is decreasing in θ and in α,

1. θd1 < θd0,

2. for any α ∈ [0, 1], bD (θ,α) ≥ bD (θ, 1) > 0 for θ < θd1,

44

3. for any α ∈ [0, 1], bD (θ,α) ≤ bD (θ, 0) < 0 for θ > θd0,

4. for θ ∈ ¡θd1, θd0¢, bD (θ,α) = 0 holds at some α = αD (θ), where ∂αD (θ) /∂θ < 0.

This completes the proof of Lemma 4.

For rF ≥ r, by the three Panels of Figure 2, there will be flipping in the uniqueequilibrium if and only if

θS (0) < θd0. (65)

By (57), θS (0) is decreasing in H, with limiting values

limH→0

θS (0) =∞ limH→1

θS (0) = 0.

By (17), θd0 satisfiesµβF

r

rF− βH (1 + z)

¶η¡θd0¢+ (1− βH)μ

¡θd0¢− (δ + r) z = 0,

whereby it is decreasing in rF . The limiting values

limrF→∞

θd0 =←−θ d0 lim

rF→rθd0 =

−→θ d0,

are the respective solutions to

−βH (1 + z) η³←−θ d0

´+ (1− βH)μ

³←−θ d0

´− (δ + r) z = 0,

(1− βH)μ³−→θ d0

´− (δ + r) z = 0,

both of which are positive and finite. Then, condition (65) cannot be met for H

close to 0 but must be satisfied for H close to 1. With θS (0) decreasing in H and θd0decreasing in rF , the combinations of H and rF under which (65) holds as an equality

define an upward-sloping relation between the two parameters. This explains the part

of Figure 3 for rF ≥ r.For rF < r, limθ→∞ bD in (63) is equal to positive infinity, whereas if the condition

in note 14 is met, ∂ bD/∂θ in (64) is at first negative, reaches zero, and becomes positivethereafter. In this case, bD, as a function of θ, is U-shaped and starts off and ends upequal to positive infinity.

Write bDmin (α; rF ) ≡ minθ

bD (θ,α) ,where by the Envelope Theorem, ∂ bDmin/∂α < 0 and ∂ bDmin/∂rF < 0. Notice also

that limrF→0 bDmin (α; rF ) = ∞ and limrF→r bDmin (α; rF ) = −∞. Then, there existsome r0 and r00 that satisfy bDmin (1; r0) = 0 and bDmin (0; r00) = 0, respectively, where0 < r0 < r00 < r.

45

For rF ≤ r0, bDmin (1; rF ) ≥ 0. Given that bD (θ,α) ≥ bDmin (α; rF ), bD (θ,α) > 0 forall α < 1. This proves Lemma 5(a) and explains the part of Figure 3 for rF ≤ r0.For rF ∈ (r0, r00), bDmin (1; rF ) < 0. Hence, there exists two θ, namely θd1 and θu1

at which bD (θ, 1) = 0, over where ∂ bD/∂θ < 0 and ∂ bD/∂θ > 0, respectively. Withinthe interval

¡θd1, θ

u1

¢, the exists some αD (θ) ∈ (0, 1) such that bD (θ,αD (θ)) = 0. For

θ /∈ £θd1, θu1¤, bD (θ, 1) > 0. Moreover, since bDmin (0; rF ) > 0 for rF < r00, αO (θ) > 0for all θ. This proves Lemma 5(b) and explains the part of Figure 3 for rF ∈ (r0, r00).At rF = r00, there is one θ at which bD (θ, 0) = 0. Call this θ∗. For θ 6= θ∗,bD (θ, 0) > 0. Then, an α = 0 equilibrium exists if and only if θS (0) = θ∗. Given that

as H varies from 0 to 1, θS (0) spans the entire positive real line, there exists one and

only one H at which θS (0) = θ∗. This explains how the U-shaped border in Figure3 is tangent to the rF = r

00 line.For rF ∈ [r00, r), bDmin (α, rF ) < 0. Then, there are two θ, namely θd0 and θu0 at

which bD (θ, 0) = 0, over where ∂D/∂θ < 0 and ∂D/∂θ > 0, respectively. Within the

interval¡θd0, θ

u0

¢, bD (θ, 0) < 0. As rF increases from r00 to r, given that ∂ bD/∂rF < 0,

and that ∂ bD/∂θ < 0 at θ = θd0 and ∂ bD/∂θ > 0 at θ = θd0, the interval¡θd0, θ

u0

¢expands, whereby ∂θd0/∂rF < 0 and ∂θu0/∂rF > 0. This proves Lemma 5(c).

As to the construction of the U-shaped border in Figure 3, first notice that in this

case, an α = 0 equilibrium exists if and only if

θS (0) ∈£θd0, θ

u0

¤.

Given that ∂θd0/∂rF < 0 and ∂θS (0) /∂H < 0, the condition θS (0) = θd0 defines an

upward-sloping border in the H-rF space. Given the continuity of bD, in the limitas rF → r, θd0 tends to the same θd0 at which rF = r just hold. Then, there is the

same H that keeps θS (0) = θd0 at rF = r. On the other hand, with ∂θu0/∂rF > 0, the

condition θS (0) = θu0 defines a downward-sloping border in the H-rF space.

Proof of Lemma 6 By (17), θu1 satisfies,µβF

r

rF− (1 + z)βH

¶η (θu1) + (1− βF )μ (θ

u1)− (δ + r) z = 0,

at where the above is increasing in θ. Let η (θ) = aθb. Then, θu1 < 1 if

βFr

rF− (1 + z)βH + 1− βF − (δ + r)

z

a> 0.

The second inequality of (23) follows under (24), which serves to ensure that rF < r.

By the proof of Lemmas 4 and 5 above, r00 satisfies,

minθ

n³βFr

r00− (1 + z)βH

´η (θ) + (1− βH)μ (θ)− (δ + r) z

o= 0.

Evaluating the above and solving for r00 yield the far left term in (23).

46

Proof of Lemma 7 Only two of the four equations of motion constitute indepen-

dent restrictions. By utilizing (1)-(4), we can reduce the system to that of two state

variables, the equations of motion of which are given by, respectively,

·nM = η (H − nM)− δnM , (66)

·nR = αδnM − η (nR − (1−H)) , (67)

and an equation for θ given by

θ =nR

H − nM . (68)

Write the Hamiltonian of maximizing (25) subject to (66)-(68) as

H = e−rt (nMυ − nRq)+ΓnM (η (H − nM)− δnM)+ΓnR (αδnM − η (nR − (1−H))) ,(69)

where ΓnM and ΓnR are the respective co-states for nM and nR. Then, we have

∂H

∂α= ΓnRδnM , (70)

·ΓnM = −e−rtυ − ΓnM

µ−η − δ + θ

∂η

∂θ

¶− ΓnR

µαδ − θ

∂η

∂θ

nR − (1−H)H − nM

¶, (71)

·ΓnR = e

−rtq − ΓnM∂η

∂θ+ ΓnR

µη +

∂η

∂θ

nR − (1−H)H − nM

¶. (72)

The last two equations imply that the co-states must be growing at the rate −rin the steady state. We can then write

ΓnM =eΓnMe−rt, (73)

ΓnR =eΓnRe−rt, (74)

for some eΓnM and eΓnR that are stationary over time. Substitute (73), (74) and thesteady-state values of nM and nR from (53) and (56), respectively, into (71) and (72),

reΓnM = υ − eΓnM µη + δ − θ∂η

∂θ

¶+ eΓnRαµδ − θ

∂η

∂θ

¶,

reΓnR = −q + eΓnM ∂η

∂θ− eΓnR µη + ∂η

∂θα

¶,

the solutions of which are

eΓnM = υ

¡r + η + ∂η

∂θα¢υ − α

¡δ − ∂η

∂θθ¢z¡

r + η − ∂η

∂θθ + ∂η

∂θα+ δ

¢(r + η)

,

eΓnR = υ

∂η

∂θ− ¡r + δ + η − θ ∂η

∂θ

¢z¡

r + η − θ ∂η

∂θ+ ∂η

∂θα+ δ

¢(r + η)

.

By (70), the optimal value for α depends on the sign of eΓnR above. This explainsE (θ) in (27).

47

Proof of Lemma 9 By (57),

1−HδH

=θS (0)

δ + η (θS (0))≤ θ

δ + η (θ), (75)

since θ ≥ θS (0). Substituting (2) into (29) and then from (59) and (53),

TV = αδnM + (H − nM) η = ηδH

δ + η(1 + θ)− η (1−H) . (76)

Differentiating and simplifying,

∂TV

∂θ= δH

µ∂η

∂θ

δ (1 + θ)

(δ + η)2− ∂η

∂θ

1−HδH

+η

η + δ

¶≥ δH

µ∂η

∂θ

δ (1 + θ)

(δ + η)2− θ∂η/∂θ

δ + η+

η

η + δ

¶> 0,

where the first inequality is by (75) and the second inequality by the concavity of η.

But then ∂θS/∂α > 0; hence ∂TV/∂α > 0.

Proof of Lemma 10 Substituting from (2), (76), and (53) and simplifying, (30)

becomes µ(1 + θ) η − 1−H

δHη (δ + η)

¶−1.

Differentiating with respect to θ yields an expression having the same sign as

−µη + θ

∂η

∂θ− 1−H

δH

∂η

∂θ(δ + 2η)

¶< −

µη + θ

∂η

∂θ− θ

δ + η

∂η

∂θ(δ + 2η)

¶=

−µ

η

δ + η

µδ + η − θ

∂η

∂θ

¶¶< 0,

where the first inequality is by (75) and the second by the concavity of η. The Lemma

follows given that ∂θS/∂α > 0.

Proof of Lemma 11 Substituting from (5), (6), and then (1), (31) becomes

1

μ+1− α

η=1− nMδnM

,

a decreasing function of nM . But where ∂nM/∂α > 0, there must be a smaller average

TBM.

Proof of Lemma 12 Setting max {VR + pFB, VU} = VU in (10) and solving (8)-

(13) for the three prices and three asset values yield the solutions in the first part of

the Lemma. Setting max {VR + pFB, VU} = VR + pFB before solving (8)-(13) yield

the solutions in the second part.

48

Proof of Lemma 13 In case D > 0, by the second part of Lemma 11, SRF has the

same sign as

(1 + z) (βHη + r) > 0.

If D = 0, by the first part of Lemma 11, SRF has the same sign as

r + (1 + z)βHη − μ (1− α) (1− βH) + (δ + r) z. (77)

But if D = 0, by (7),µβF

r

rF− βH − zβH

¶η + (1− βH − α (βF − βH))μ = (δ + r) z.

Substituting into (77) for (δ + r) z,

(rF + βFη) r + (1− βF ) rFμα

rF> 0.

If D ≤ 0, by the first part of Lemma 12,

SRU =υ

r + δ + βHη> 0.

This completes the proof of Part (a) of the Lemma.

For Part (b), if D ≥ 0, the condition to check is

VR + pFB ≥ 0.

By the second part of Lemma 12, the condition becomes

r

rFβFη + (1− βF )μ− (δ + r) z ≥ 0,

which holds whenever bD ≥ 0. If D ≤ 0, we want to show thatVU ≥ 0,

which holds by the first part of Lemma 12.

Proof of Proposition 1 Part (a) is obtained by setting bD (θ,α) and E (θ), givenby (17) and (27) respectively, equal to each other and solving for βH and βF . For

part (b), if ∂ bD (θ,α) /∂θ > 0,(1− βH − α (βF − βH)) >

µβF

r

rF− βH − zβH

¶ ∂η

∂θθ2

η − θ ∂η

∂θ

> 0.

49

Then,

bD (θ,α)−ES (θ) =

µβF

r

rF− βH − zβH

¶η + (1− βH − α (βF − βH))

η

θ

−∂η

∂θ+

µη − θ

∂η

∂θ

¶z.

>

µβF

r

rF− βH − zβH

¶Ã1

1− ∂η

∂θθη

!+

µ1− θ

η

∂η

∂θ

¶z − ∂η

∂θ

1

η

> 2

∙µβF

r

rF− βH − zβH

¶z

¸1/2− ∂η

∂θ

1

η.

The second inequality comes from minimizing

βFrrF− βH − zβH

ε+ zε

with respect to ε. Rewriting the last line gives the condition in the Proposition.

Proof of Proposition 2 We begin with showing search market price pFS in the

full-intermediation equilibrium, given by (34), is higher than pFS in a partially-

intermediated equilibrium, given by (36). Now, at where θ = θu1 , D (θ, 1) = 0, the

two pFS are by construction equal. Second, with pFS in the first equation increasing

in θ by the concavity of η and the pFS in the second equation increasing in θ for

rF < r, which is a necessary condition for multiplicity, pFS in the full-intermediation

equilibrium must exceed pFS in the partially-intermediated equilibrium, since in this

case θS (1) ≥ θu1 whereas pFS in the partially-intermediated equilibrium has a θ ≤ θu1 .

Lastly, with pFS > pH in the partially-intermediated equilibrium, the single search

market price in the full-intermediation equilibrium exceeds the two search market

prices in the partially-intermediated equilibrium. Next, in a comparison between pFSin two partially-intermediated equilibria, given that pFS in (36) is increasing in θ,

there must be higher pFS in the larger θ equilibrium. The same ranking applies to

the two pH , given that pH in (37) is similarly increasing in θ in case rF < r. The

final comparison is between pH in a partially-intermediated equilibrium and pH in

the no-intermediation equilibrium, given by (32). At where θ = θu0 , D (θ, 0) = 0, the

two pH are by construction equal. With the first pH known to be increasing in θ, the

proof is completed by noting that pH , given by (32), is likewise is increasing in θ given

the concavity of η. This completes the proof that search market housing prices across

steady-state equilibria can be ranked by the value of θ. Given that pFB =η

η+rFpFS,

investment market housing prices are ranked in the same order as in search market

housing prices.

Proof of Proposition 4 By differentiating (38)-(40) with respect to θ and noting

that αS ≤ 1 and ∂αS/∂θ > 0.

50

Proof of Proposition 6 Define Φ (α) ≡ αO (θS (α)), a continuous function map-

ping [0, 1] into itself. A steady-state equilibrium is any fixed point of Φ. By Brouwer’s

Fixed Point Theorem, a continuous function mapping the unit interval into itself must

possess a fixed point.

Proof of Proposition 7 Write R = r/rF , which remains constant amid any

equiproportionate changes in r and rF . Substituting rF = rR−1 into (33) and (34)

and differentiating proves the first part of (a). In a no-intermediation equilibrium, pHis given by (32), which is independent of rF but decreasing in r for sufficiently small

θS (0) and/or q. This proves the second part of (a). For (b), substituting rF = rR−1

into (35)-(37), respectively, yields,

pFB =βFη

rR−1 (r + δ + βHη)υ, (78)

pFS =βFη + βF rR

−1

rR−1 (r + δ + βHη)υ, (79)

pH =βFη + βHrR

−1

rR−1 (r + δ + βHη)υ. (80)

all of which are decreasing in r. Solving bD = 0 from (17) for

r =(βFR− βH − zβH) η + (1− βH − α (βF − βH))μ

z− δ, (81)

and substituting into (78)-(80), respectively, gives

pFB =RβFηq

2

G, (82)

pFS =((βFR− βH) η + (1− βH − α (βF − βH))μ)βFυ − (ηβH + δ − ηR)βF q

Gq,

(83)

pH =((βFR− βH) η + (1− βH − α (βF − βH))μ) βHυ − (βH (ηβH + δ)− βFηR) q

Gq,

(84)

where

G = (((βH − βFR) η − (1− βH − α (βF − βH))μ) υ + (δ + βHη) q) (85)

× ((βH − βFR) η − (1− βH − α (βF − βH))μ) .

Differentiating (82) with respect to θ, evaluating the resulting expression at βF =

βH = 1/2 yields an expression whose sign is given by that of

η (R− 1 + 1/θ − z)− 2δz − (R− 1 + 1/θ)µθ2∂η

∂θ(R− 1 + 1/θ − z)− η

¶.

51

The expression is strictly positive at R = 0 and R = 1 + z if the RHS of (81) at

βF = βH = 1/2 is positive. And then differentiating twice with respect to R yields

−2θ2∂η∂θ< 0.

Thus, pFB in (82) must be increasing in θ for R ∈ [0, 1 + z]. For pFS and pH ,

differentiating (83) and (84) with respect to θ and evaluating at βF = βH = 1/2 and

R = 0, 1, and 1+ z, respectively, all yield a strictly positive expression as long as the

RHS of (81) is positive at βF = βH = 1/2. Then, pFS and pH in (83) and (84) must

be increasing in θ for R in neighborhoods of 0, 1, and 1 + z.

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